With many of its own members at elevated health risk from the virus and bracing for potential travel restrictions, the Senate may have only a handful of days to sign a deal before leaving town for what could be an extended recess. And though the House is not in session and has not scheduled a return to Washington, many of its members were ready to fly back if there were a deal.

Senator Mitch McConnell, Republican of Kentucky and the majority leader, said his party was coming close to an agreement with the White House and could begin talks as early as Thursday with Democrats, whose support would be needed to pass anything into law.

“Republicans hope shortly to have a consolidated position along with the administration, then we intend to sit down with our Democratic colleagues to see what we can agree to,” he said Wednesday afternoon. “Just how long it will take to get through these steps is unclear, but as everyone knows, we are moving rapidly because the situation demands it.”

In a conference call with Democratic leadership and committee chairmen on Wednesday afternoon, Speaker Nancy Pelosi said she was pushing for a bipartisan negotiation between House and Senate leaders that could yield a final agreement on the next economic rescue package by early next week.

The Trump administration’s proposed relief package, outlined in a document circulated to lawmakers and lobbyists, called for the authority to send two waves of checks, each totaling $250 billion, directly to American taxpayers, the first on April 6 and the second on May 18. Payments would be fixed and their sizes dependent on income and family size, the summary said, in line with a proposal by Senator Josh Hawley, Republican of Missouri.

It would extend loans to small businesses equaling six weeks of their payroll up to $1,540 per employee, on the condition that the companies must keep paying their employees for eight weeks after receiving the loan. It also called for $50 billion for secured loans for the airline industry and another $150 billion for secured loans or loan guarantees for other sectors of the economy that have been devastated by the global economic shutdown as the virus spreads.