Ten Republican senators wrote to President Biden on Sunday, outlining a framework for coronavirus relief legislation and pushing for Mr. Biden to compromise. The letter came as Democrats were preparing to bypass the need for Republican support in order to deliver a sweeping relief package.

The 10 senators, led by Susan Collins of Maine, proposed a framework that includes some of the provisions in Mr. Biden’s $1.9 trillion plan but would curtail others. It would include $160 billion for vaccine distribution and development, Covid testing and the production of personal protective equipment, along with relief for schools, small businesses and “more targeted assistance” for individuals through another round of direct payments and unemployment benefits.

“Our proposal reflects many of your stated priorities, and with your support, we believe that this plan could be approved quickly by Congress with bipartisan support,” the senators wrote.

The group, which also includes Mitt Romney of Utah, Michael Rounds of South Dakota, Lisa Murkowski of Alaska and Shelley Moore Capito of West Virginia, is expected to release additional details of their proposal on Monday.

To pass a package through the regular legislative process, Mr. Biden would need 60 votes in the Senate, and therefore the support of at least 10 Republicans.

Mr. Biden and top Democrats have said they want Republican support for a new relief bill. But with several Republicans already balking at passing a sweeping package, Democrats are preparing to pass a bill on their own using budget reconciliation, a legislative process that is more complicated but requires only a simple majority in the Senate.