Negotiations have been further complicated by the presence of thousands of Central American migrants on the border between Mexico and the United States. After the migrants tried to evade a police blockade and rush across the border, Border Patrol agents fired tear gas on the group — including women and children — to force them to turn back.

The altercation has hardened stances on both sides of the debate, fueling Democrats’ outrage over the use of force and solidifying Republicans’ argument that Mr. Trump’s promise of a wall was vital to his pledge to protect the country — and that Democrats are standing in the way.

“Our Democratic colleagues seem to come up with a lot of criticism, but have no answer when you ask the question, ‘What are we supposed to do about people breaking through barricades, mobs of people coming into our country illegally and not following the rules?’” Senator John Cornyn of Texas, the majority whip, said during a weekly leadership conference.

Still, it has been Mr. Trump who has been spoiling for a shutdown fight over the wall for more than a year, saying last year that a “good shutdown” might be needed to bring Congress to heel on the issue, and telling reporters this month that “this would be a very good time to do a shutdown.”

As the impasse continued, lawmakers on both sides of the Capitol were also contemplating resurrecting the long-abandoned idea of a deal in which Democrats would agree to Mr. Trump’s wall-funding demands in exchange for a pathway to legal status for a group of young immigrants who were brought to the United States illegally as children.

“I think there’s easily a deal to be made there on the border wall and DACA, if people want to make it,” Senator Roy Blunt, Republican of Missouri, said on Tuesday.

The president sought to strike such a trade last year with Democratic congressional leaders after he rescinded Deferred Action for Childhood Arrivals, or DACA, a program President Barack Obama created in 2012 to shield the young immigrants from deportation and grant them work permits. But it collapsed after months of haggling, with Republicans insisting it must be accompanied by stricter immigration measures and Democrats split on agreeing to any such compromise.