The president “has given Democrats a great opportunity to show how we will govern responsibly & quickly pass our plan to end the irresponsible #TrumpShutdown,” she said, “just the first sign of things to come.”

Democrats also intend to use their first months in the majority to push for a bipartisan infrastructure bill and legislation to lower prescription drug costs, issues that they believe will have bipartisan appeal.

The Democrats plan to pass two bills on Thursday. The first includes six bipartisan spending measures that would fully fund agencies like the Interior Department and the Internal Revenue Service through the end of the fiscal year in September. The second would fund the Department of Homeland Security through Feb. 8, with $1.3 billion for fencing but no money for a wall on the Mexican border.

With the plan facing a shaky future in the Senate and an intransigent president, some rank-and-file lawmakers in both parties are suggesting that a deal to revamp the nation’s immigration laws, pairing border security and protections for some undocumented immigrants, may be the way out of the stalemate.

“How about comprehensive immigration reform?” Representative Debbie Dingell, Democrat of Michigan, said on CNN on Tuesday, when asked how Democrats intend to compromise with Mr. Trump.

One of Mr. Trump’s closest allies, Senator Lindsey Graham, Republican of South Carolina, has revived his long-stalled immigration proposal to marry $5 billion for the wall with immigration law changes that might appeal to Democrats, including three-year renewable work permits for young undocumented immigrants brought to the country as children, known as Dreamers.

Mr. Trump has raised the prospects of broader talks on Twitter. “We will be forced to close the Southern Border entirely if the Obstructionist Democrats do not give us the money to finish the Wall & also change the ridiculous immigration laws that our Country is saddled with,” he said last week. “Hard to believe there was a Congress & President who would approve!”