The most immediately critical legislation may be the seven remaining appropriations bills to ensure that the entire government remains funded past a Dec. 7 deadline. Issues including a fight over spending on border security — the Senate included $1.6 billion in its bill, compared with the House’s $5 billion — have bogged down negotiations, and Mr. Trump has threatened to veto legislation without funds for a wall at the border with Mexico.

“We have the will to put the money at the border for better security and combine it with some sensible reforms, including things like a path to citizenship, things like making sure that we have workers on our fields and in our factories that we need,” Senator Amy Klobuchar, Democrat of Minnesota, said on ABC’s “This Week.” “But what I don’t think we should do is shut down the government. And that, again, is in his hands and his party’s hands.”

Lawmakers from both parties have said that they want to avoid a shutdown, and that passing the appropriations bills — which would fund a number of federal agencies, including the Homeland Security, Agriculture and Commerce Departments — is a priority for their final weeks. But any resolution needs at least some bipartisan support to clear the Senate’s 60-vote threshold, leaving a resolution unclear. Aides involved with the negotiations said a possible path forward might become apparent only in the coming days.

If negotiations falter, Congress could avoid a shutdown by temporarily funding the government into the new year, but that would force it to take up the appropriations fight again in January. Lawmakers have already passed spending measures covering other parts of the government.

And Mr. Trump remains a wild card. Despite attempts from Republican leaders to dissuade him, he doubled down on his veto threat during his Thanksgiving vacation in Florida.