But the proposal, which Mr. Trump unveiled in a 13-minute address from the White House, appeared dead on arrival in the Capitol. Speaker Nancy Pelosi rejected it even before Mr. Trump spoke, and Senator Chuck Schumer, the Democratic leader, denounced the offer as “not a compromise but more hostage taking.”

With the shutdown entering its fifth week and polls showing a majority of the public blaming Mr. Trump, the president’s advisers have been searching for an exit strategy.