One question for this week is how many Republicans will vote for the president’s offer. Mr. Trump’s previous immigration proposal last year failed in the Senate, when 14 Republicans voted against it. In the Senate, casting votes on legislation has traditionally helped spark negotiations, though bipartisan groups of senators have already been working on compromise proposals for weeks, none of which have secured Mr. Trump’s support.

One Democratic aide said Mr. Trump’s offer could bolster his case to congressional Republicans that he has done all he can to reopen the government, leaving him no option but to declare a national emergency. White House chief of staff Mick Mulvaney told reporters on Saturday that declaring a national emergency, which may let the president spend money on a border without congressional approval, was “still a tool that’s available to the president.”

But asked about what would happen if there was opposition from Democrats, much of Mr. Trump’s negotiating team described the legislative proposal as their best offer to reopen government.