By Natalie Andrews and Lindsay Wise, Wall Street Journal–

Disagreements emerged among senators over how to help individual Americans during the coronavirus pandemic as lawmakers raced to meet a midnight deadline Friday to reach agreements on a wide-ranging stimulus package.

Lawmakers debated whether to give cash payments directly to Americans, expand unemployment insurance or some combination of the two. A Senate Republican opening offer and the Trump administration called for direct payments to workers, while Democrats have pushed for dramatically increasing unemployment benefits.

“Senate Republicans want to put cash in Americans’ hands—for Americans who’ve lost work this would be the quickest possible first wave of government assistance to supplement unemployment insurance and help with bills and immediate needs,” Senate Majority Leader Mitch McConnell (R., Ky.) said on the Senate floor.

Democrats want to expand jobless benefits, arguing their plan would help more people for a longer time and include contract and gig workers.

“It makes the most sense of any program that I’ve heard in dealing with those who are losing their jobs and in terms of dealing with stimulating the economy, because we’ll bring the economy back to where it was,” Sen. Chuck Schumer (D., N.Y.), the Democratic leader, said of expanding unemployment insurance.

Mr. McConnell set a midnight Friday deadline so that the legislation could be drafted and voted on by Monday, a breakneck pace for the chamber known for deliberating.

The bipartisan groups of senators met Friday to discuss plans for people and industries facing the brunt of the struggling economy as Americans stay home and away from businesses and travel, with several administration officials, including Treasury Secretary Steven Mnuchin in attendance. Any eventual deal will need to be bipartisan because it requires 60 votes in the Senate and must pass the Democratic-controlled House to become law, meaning Republicans and Democrats will have to bridge their divisions.

“There’s tremendous spirit to get something done,” said President Trump at the daily White House coronavirus briefing. He said he had good telephone calls with Messrs. Schumer and McConnell.

While sounding optimistic notes about the state of talks, Mr. Trump also bristled at suggestions that the government’s effort was falling short on testing, potential treatments and medical equipment. When a reporter asked what the president would say to reassure Americans who are sick or scared, Mr. Trump responded: “I think it’s a very nasty question, and I think it’s a very bad signal that you’re putting out to the American people.”

Democrats are pushing for a “Marshall Plan” for the nation’s health-care system, calling for a $400 billion federal investment to address capacity issues and medical-supply shortages at hospitals, particularly rural and smaller ones.

“Our hospitals across the country are about to be overwhelmed,” said Illinois Sen. Dick Durbin, the No. 2 Senate Democrat. “They’ll be faced to make some horrible triage choices about survivors. We don’t want to reach that point.” He called for a doubling of hospital capacity.

Mr. McConnell’s broad plan, released late Thursday, calls for taxpayers to receive up to $1,200, with married couples eligible to receive as much as $2,400 with an additional $500 for every child. Those payments will scale down for individuals who make more than $75,000 a year and couples that make more than $150,000. Individuals who make more than $99,000 and households that earn more than $198,000 won’t be eligible for direct assistance.

But lawmakers in both parties were pushing Friday for a number of changes or additions to that proposal. Members of both parties expect that expanding unemployment insurance will be included in the final deal in addition to some sort of cash assistance.

“What we as Republicans got to understand: There’s got to be an unemployment compensation benefit part of the package, and I support that. It makes sense to me. Everybody’s not going to be able to be run through these systems,” Sen. Lindsey Graham (R., S.C.) said.

Sen. John Thune (R., S.D.) said “we’ll see” whether changes will be made to the direct-payments proposal in the GOP plan, saying there are “real concerns about the alternative was backfilling money through unemployment insurance accounts, state accounts, and there are real concerns about whether or not that can be done.”

Administration officials have also raised concerns about how quickly the existing unemployment insurance program could disburse funds to Americans.

Republicans and Democrats also have both criticized how Mr. McConnell’s plan for cash assistance treats low-income people. Under the Senate GOP opening proposal, Americans who didn’t make enough money to pay income tax would only be eligible for up to $600. Republicans and Democrats want to boost assistance to those low-income Americans.

“I couldn’t believe that they were talking about lowest-income people getting $600 and somebody making $75,000 getting twice as much as that, $1,200. So those numbers don’t make any sense,” said Sen. Debbie Stabenow (D., Mich.).

Under the Republican proposal, the government will provide $50 billion in loan guarantees for passenger air carriers, $8 billion for cargo air carriers and $150 billion for other large businesses, authorizing the government to take equity stakes in them. The proposal also includes $300 billion for loan guarantees to small businesses.

Mr. McConnell’s plan didn’t include additional funding for federal agencies that have been strapped for cash during the crisis. The Trump administration is seeking $45.8 billion in additional funding for agencies like the Centers for Disease Control and Prevention and the Defense Department.

Sen. Richard Shelby (R., Ala.) said that additional funding for federal agencies may also be included in the final agreement.

Beyond the direct payments to individuals, the Republican proposal would let employers defer payroll tax payments until 2021 and 2022. Companies would also get more flexibility to use operating losses against past profits to get quick refunds, and businesses would get reprieves from several errors or ambiguities in the 2017 tax law that they have been complaining about for years.

On education, the Republican plan would allow the secretary of education to defer student loan payments for three months without a penalty.

The GOP plan would also relax restrictions on telehealth, expand flexible- spending accounts to pay for certain over-the-counter drugs, and boost Medicare reimbursements to providers.

House Democrats aren’t scheduled to come back to Washington until there is legislation to vote on, though House Speaker Nancy Pelosi (D., Calif.) is in close contact with Mr. Schumer on the matter, aides said.

In a suggestion likely to anger Democrats, Republicans are also seeking to roll back the Affordable Care Act employer-coverage mandate, which requires employers with 50 or more full-time workers to offer health insurance or pay a penalty. Republicans say a rollback would provide relief for some businesses like grocers that might need to temporarily increase staffing.

In a Congress typically beset by partisan polarization and legislative gridlock, lawmakers have advanced legislation with unusual speed, passing an initial $8.3 billion bill that included funding for vaccine development and then passing this week subsequent legislation providing for paid leave that is estimated to cost more than $100 billion.

Write to Natalie Andrews at Natalie.Andrews@wsj.com and Lindsay Wise at lindsay.wise@wsj.com