After weeks of closures and social-distancing orders in the U.S., states from Mississippi to Tennessee to Colorado began to permit some businesses to reopen Monday, welcoming customers back and letting some employees return to work. Over the weekend, some businesses had resumed in Georgia, Oklahoma, Alaska, Texas and South Carolina, with social-distancing measures in place.

Retail stores, restaurants, malls, movie theaters, museums and libraries in Texas will be allowed to open Friday at 25% capacity, Republican Gov. Greg Abbott said Monday. He will allow a stay-home order now in place to expire Thursday.

“The executive order has done its job,” he said. “Now it is time to start a new course.”

Many are watching Texas’s phased reopening as a test. The country’s second most populous state has relatively low rates of Covid-19, but among the lowest testing rates, leading critics to say there is no way to know how prevalent the virus really is.

Public-health officials have cautioned that lifting restrictions hastily could lead to a surge in new cases. Health experts say states shouldn’t reopen unless they keep new daily infections below established thresholds, show capacity for expanded testing and have public-health workers ready to track new infections.

Trump administration guidelines, which lay out three phases of reopening, place the onus on governors to lift measures based on state data. The administration plans to send all 50 states supplies to test at least 2% of residents, The Wall Street Journal reported.

President Trump emphasized the administration’s testing plans during a teleconference with governors Monday, saying the U.S. was making “encouraging progress” and praised state leaders for their efforts to expand testing. The Journal listened to an audio recording of the call.

“As the rate of new cases continues to decline we’re glad to hear that more than half of our nation’s governors have announced plans to begin a safe phased opening,” Mr. Trump said. “People want to see these things open.”

Attorney General William Barr warned Monday that state and local coronavirus-related restrictions would be closely monitored, directing federal prosecutors to be on the lookout for policies that could violate constitutional rights. Mr. Barr had signaled last week that the Justice Department would support some legal challenges to state’s lockdown orders.

Mr. Barr, who has been a vocal champion of religious freedom, cited policies that discriminate against religious institutions, “disfavored speech” and “undue interference with the national economy” as potential examples.

“Many policies that would be unthinkable in regular times have become commonplace in recent weeks, and we do not want to unduly interfere with the important efforts of state and local officials to protect the public,” Mr. Barr wrote in a memo to U.S. attorneys. “But the Constitution is not suspended in times of crisis. We must therefore be vigilant to ensure its protections are preserved, at the same time that the public is protected.”

He tapped the head of the Justice Department’s civil-rights division and the top federal prosecutor in Detroit to oversee the effort.

Confirmed coronavirus cases world-wide Monday topped 3 million, with more than 210,000 deaths, according to data from Johns Hopkins University.

The U.S. accounted for about a third of global infections, with more than 983,000 confirmed cases and more than a quarter of reported deaths, at 55,735, according to the Johns Hopkins data. Some experts say the figures understate the extent of the pandemic.

Infections continued to rise in Massachusetts with nearly 55,000 confirmed cases reported, the third-highest of any state, and nearly 3,000 deaths. Gov. Charlie Baker urged residents to stay home and continue practicing social distancing. “We here in Massachusetts, as I said earlier in my remarks, are still in the surge and very much in the fight against Covid-19,” the Republican governor said.

Louisiana Gov. John Bel Edwards, a Democrat, extended the state’s stay- at-home order until May 15, acknowledging that the number of new cases and new hospitalizations have continued to go up in some parts of the state.

In the worst-hit states of New York and New Jersey, governors are still reporting a sizable number of new infections and deaths, though they have pointed to signs of the virus slowing. New York Gov. Andrew Cuomo has outlined a phased reopening for the state, centered on Centers for Disease Control and Prevention guidelines that permit reopenings after a 14-day drop in the state’s hospitalization rate.

Mr. Cuomo, a Democrat, said Monday he would likely extend lockdown orders, set to expire May 15, in parts of the state most affected by the virus, including New York City.

The New York State Board of Elections, meanwhile, canceled the state’s Democratic presidential primary, which was scheduled for June 23. And New York City Mayor Bill de Blasio said the city planned to hire 1,000 contact tracers.

In New Jersey, Democratic Gov. Phil Murphy outlined benchmarks for the state before it begins a phased reopening, including a sustained drop in Covid-19 cases and hospitalizations over 14 days, doubling the state’s testing capacity and securing space to isolate and quarantine future patients.

“This isn’t just about New Jersey,” Mr. Murphy said. “For us to rush ahead of either Pennsylvania or New York, or any of our other four state partners—or vice versa—would risk returning our entire region back into lockdown mode.”

“We only want to have to do this once,” he added.

Governors have taken different approaches to reopening, though state leaders in the Northeast, Midwest and West have formed regional coalitions to coordinate reopening plans. Colorado and Nevada joined the West Coast pact Monday.

In California, public-health officials in six Bay Area counties and Oakland announced they would extend shelter-in-place orders through May, though some lower-risk activities would be able to resume.

Democratic Gov. Gavin Newsom criticized beachgoers in Southern California who took advantage of summerlike weather over the weekend to hit the sand, saying disregarding social-distancing guidelines could delay reopening procedures and ruin progress made against the virus.

“This virus doesn’t take the weekends off,” said Mr. Newsom, who added that the state was weeks away from making “measurable” changes to its stay-at-home order.

Missouri, meanwhile, will enter the first phase of its recovery plan May 4 as businesses gradually reopen and residents can return to social activities, so long as they maintain social distancing precautions. GOP Gov. Mike Parson said it was possible some communities would reopen sooner than others.

In Ohio, Gov. Mike DeWine said retailers would be able to reopen in mid-May, so long as customers and employees wear masks and follow other protective guidelines. “With any decision, there is risk,” the Republican said.

Stocks rose Monday, as investors expect stimulus measures and an easing of lockdown measures in the U.S. and world-wide to help kick-start economic activity. Meanwhile, companies of all stripes are scrambling to avoid painful reorganizations of their capital structures and operations, default or bankruptcy.

In the U.K., Prime Minister Boris Johnson, back at work after recovering from a serious coronavirus infection, said the country was passing through the peak of the pandemic, and urged people to keep respecting lockdown measures. The government would soon outline a plan to start easing restrictions as the virus starts to wane, he said, adding that the country was now in the period of “maximum danger.”

New Zealand said it had effectively eliminated the pathogen from within its borders, as one new case was confirmed Monday. “There is no widespread undetected community transmission in New Zealand,” Prime Minister Jacinda Ardern said. “We have won that battle.”

Ms. Ardern warned New Zealanders to remain vigilant and left the door open for restrictions to be tightened once more if infection rates rise again.

Hard-hit Italy on Sunday announced a timetable for reopening, with Prime Minister Giuseppe Conte saying restrictions will be lifted in stages starting May 4, provided social interactions be kept at a distance of just over 3 feet. “If you love Italy, keep your distance,” Mr. Conte said.

India’s government in recent days said many shops could reopen while malls must remain closed. The country imposed a nationwide lockdown in March that is set to lift May 3.

India and other developing countries that have relaxed restrictions—including Pakistan, Ghana and South Africa—are encouraged by the relatively low number of infections they have seen compared with hot spots in China, southern Europe and the U.S.

But health specialists worry that the case numbers may underestimate the extent of infections and these countries might be in the early stages of an epidemic that arrived later than in the developed world.

China reported three new coronavirus cases—a record daily low—roughly four months after the virus began emerging there. The government has eased lockdowns but is struggling to get the economy back on track.

Industrial profits in China fell 34.9% in March from a year earlier, the National Bureau of Statistics said Monday. That was a slower pace of decline than the 38.3% drop in the first two months of the year, but shows the severity of disruptions caused by the pandemic.

Japan reported the fewest cases in a month as the state of emergency declared in Tokyo and other cities on April 7, and subsequently extended nationwide, began to show its impact. The government said Monday that 203 new cases were reported nationwide Sunday.

Hong Kong and Taiwan both reported no new coronavirus cases Monday. It was the second day in a row that Hong Kong recorded zero new infections. Taiwan has reported no local transmissions since April 12.

Corrections & Amplifications
Missouri’s stay-at-home measure will remain in place through May 3, and some Tennessee businesses reopened their doors today. An earlier version of this article incorrectly said Missouri had begun to ease its lockdown this week and that some businesses in Tennessee reopened over the weekend.

Write to Jennifer Calfas at Jennifer.Calfas@wsj.com, Elizabeth Findell at Elizabeth.Findell@wsj.com and Newley Purnell at newley.purnell@wsj.com