Republicans have also expressed strong opposition to adding money for states and municipalities, saying Democrats have pushed for unrestricted funds, not related to the coronavirus, that would effectively subsidize bad fiscal decisions that occurred before the pandemic. That has been a red line for Republicans throughout the talks.

But after the funding for the Paycheck Protection Program lapsed, Republicans expressed the first hints of openness to accepting at least some of the Democrats’ demands. In an interview with Politico on Friday, Mr. McCarthy said he was “fine with doing some hospital” funding as part of a package to shore up the program.

Some Republicans, though, have expressed skepticism about Mr. Mnuchin, whom they see as accommodating to Democrats. Asked on Thursday about how a deal that included hospital money would be received, Senator Mitch McConnell, Republican of Kentucky and the majority leader, remained noncommittal, saying only, “We’d take a look at it.”

Mr. Mnuchin also said on Sunday that he was hopeful that the economy could rebound in a matter of months rather than years. He said that he hoped the extraordinary efforts the government had taken to encourage businesses to keep workers on their payrolls would prevent the jobless rate from reaching 20 percent.

Mr. McConnell hosted a call on Sunday with Mr. Mnuchin, Mr. Trump, Republican senators and Mark Meadows, the White House chief of staff, about the ongoing negotiations. Mr. McConnell said that additional funds for state and local governments, as well as more money for food assistance, would not be included in the final package, according to an aide for a Republican leader who requested anonymity to disclose details of a private phone call.

Mr. Mnuchin added that some of the unresolved items were related to funding for testing, and that he would be conferring with Mr. McConnell, Senator Lamar Alexander of Tennessee, the chairman of the Senate health committee, and Senator Roy Blunt of Missouri, the chairman of the Senate Rules Committee, to resolve the issues to ensure swift passage.