Senate Republican leaders have signaled that their proposal for the next coronavirus relief package will include additional funding to test students for the coronavirus.

“The most important thing we need for normalcy is to get people back into school,” Senator Roy Blunt, the Republican chairman of the Senate appropriations subcommittee that handles funding for education and health, told reporters Tuesday. “And we’re not going to do that particularly in a residential setting without millions of tests people can take dozens of times.”

The Republican proposal, which Blunt said would likely be unveiled in about a month, will include more money for testing nationally and to ensure enough funding to develop a vaccine.

But laying out a partisan divide in the Senate over how much additional help the next package will include for higher education, the Senate’s Democratic leader, Chuck Schumer, and Patty Murray, the top Democrat on the education committee, jointly proposed a broader $430 billion coronavirus aid relief package for childcare and education.

The proposal would give colleges and universities an additional $132 billion.

Terry Hartle, the American Council on Education’s senior vice president for government and public affairs, said the package is more comprehensive than what Republicans are expected to propose. While Republicans are highlighting help for colleges and schools to reopen, the Democratic proposal would go beyond the $74 billion institutions estimate it will cost them to resume in-person instruction. It would also cover the $46 billion colleges and universities say they need to cover losses during the pandemic.

“A number of Republicans are interested in the link between additional federal support and safely reopening critical parts of our economy, including colleges and universities,” said Craig Lindwarm, vice president of congressional and governmental affairs at the Association of Public and Land-grant Universities.

In addition, the Senate’s third-highest-ranking Republican, John Barrasso, illustrated another key division between Republicans and Democrats in being able to reach a deal on a new relief package. Barrasso, at a news conference, repeated the Republicans’ insistence that any package include liability protection for companies and colleges from coronavirus-related lawsuits.

Meanwhile, during an eventful day in which Dr. Anthony Fauci and other top federal health experts testified before a Senate committee, the Centers for Disease Control and Prevention also updated its nonbinding guidance on whom colleges and universities should test.

Predictably, the agency said colleges should refer those with symptoms to a health-care provider to evaluate whether they should be tested, and should separate those who show symptoms from others.

The CDC also took a position on a question that has divided colleges — whether to test all students and faculty members as they return to campus, regardless of whether or not they have symptoms, because the tests are imperfect.

A Johns Hopkins University study found that even on the day before a person starts showing symptoms, tests will say that two-thirds with coronavirus but no symptoms do not have the disease. Some institutions, including the University of California, San Diego, are planning to test all students anyway, arguing that despite the imperfections, testing everybody would still identify a third of asymptomatic but infected people, who can be quarantined so they do not spread the coronavirus to others.

The CDC, though, said testing everyone is not recommended because it’s unknown if comprehensive testing is more effective than implementing other measures, such as social distancing and wearing masks.

Dr. Georges Benjamin, executive director of the American Public Health Association, agreed with the CDC.

The tests will identify some who have the virus, he said. “The problem is it’s like finding a needle in a haystack.” Those who test negative could also get the virus the next day, said Benjamin.

Paying for Tests

Meanwhile, Senator Lamar Alexander, the Republican chairman of the Health, Education, Labor and Pensions Committee, also signaled some willingness to help colleges pay for testing in the fall, after rulings by several federal agencies last week left it unclear whether tests to screen students and workers will be covered by insurance.

The Centers for Medicare and Medicaid Services, and in a separate guidance, the Health and Human Services and Labor Departments, along with the Internal Revenue Services, last week said health insurers only are required to pay for tests ordered by a doctor when a person is showing symptoms for the virus or has been in contact with someone who is sick.

During a hearing of his committee, Alexander said who pays for tests, if insurers will not, has to be cleared up.

“A school may want to do random testing. Perhaps it should make an arrangement with the state to pay for that,” he said. “Or perhaps Congress needs to provide more money to pay for that.”

Hartle, though, said colleges are planning a wide range of approaches to testing, including relying on local state health clinics. It’s also unclear how many are planning on using students’ or workers’ insurance to pay for the costs.

Alexander on Tuesday repeated his push for colleges and schools to reopen.

“I have said, to everybody that will listen, that I think it’s very important that schools and colleges have the money they need to reopen safely this fall,” he told reporters after the hearing. “I mean, nothing will move us more rapidly toward normalcy in the year 2020, before we have vaccines, than for 75 million students to go back to 135,000 schools and 6,000 colleges in the fall. So I’m looking carefully at what those needs are.”

Alexander also prodded President Trump to begin wearing masks occasionally.

“Unfortunately this simple life-saving practice has become part of a political debate that says, If you’re for Trump, you don’t wear a mask. If you’re against Trump, you do,” Alexander said. “That is why I have suggested the president should occasionally wear a mask even though there are not many occasions when it is necessary for him to do so. The president has millions of admirers. They would follow his lead. It would help end this political debate. The stakes are too high for it to continue.”

But among the political issues raised by reopening is whether the federal government should create regulations to protect workers during the pandemic. The Labor Department’s Occupational Safety and Health Administration has not issued any rules. Senator Tammy Baldwin, Democrat from Wisconsin, said that rewards bad apples and is unfair to businesses that are taking precautions to protect workers.