In the plan he unveiled on Saturday, Mr. Trump offered to restore the DACA and T.P.S. protections for three years. But Democrats say that is a nonstarter, because it does not offer a pathway to citizenship for Dreamers, and because Mr. Trump is simply giving back protections that he himself took away.

The McConnell aide, Don Stewart, said the Republican legislative package would include seven appropriations bills that would fund government agencies that have been partially closed for a month.

“The legislation that the majority leader will bring to the floor this week would both reopen the remaining portions of the government, fund disaster relief, fund border security and address immigration issues that both Republicans and Democrats would like to address — all in one bill,” Mr. Stewart said.

Mr. McConnell would need votes from all Republicans and seven Democrats for the package to pass. But only a handful of Democrats in Republican-leaning states might feel pressured enough to cross the aisle. A spokesman for one of them, Senator Joe Manchin III of West Virginia, said Mr. Manchin — who just won re-election — would remain undecided on the Republican proposal until he saw it.

In broadening the package beyond Mr. Trump’s plan, Mr. McConnell may be trying to pick up additional Democrats — especially those who are eager to vote for disaster relief.

“He’s starting to move the pieces around to see what’s going to fly and what’s not going to make it,” said Jim Manley, who was a top aide to Harry Reid, the former Democratic leader from Nevada. “He’s trying to put the pressure back on Democrats, and he’s trying to see what the possibilities of a deal are by putting a larger package there. He’s trying to see how large the package has to be to bring Democrats on board.”

But Mr. Trump seemed to be holding fast. He retweeted a series of endorsements of his proposal from Republican lawmakers on Sunday evening, and earlier in the day, he took aim at Ms. Pelosi and pushed back against conservative critics who have described the plan as amnesty for undocumented immigrants.