The details: The financing, including for 55 miles of new fencing, is far less than the $5.7 billion Mr. Trump has demanded, and not even as much as the $1.6 billion a Senate panel approved last year. The president refused that earlier legislation, leading to the longest government shutdown in U.S. history.

What’s next: The latest deal, which would still have to pass in the House and Senate, could be finalized today, well before funding is scheduled to lapse for several federal agencies on Friday. Lawmakers expressed confidence that Mr. Trump would be willing to sign the measure.

Reaction: Mr. Trump was informed of the agreement shortly before appearing at a rally in the border city of El Paso on Monday night. “We’re building the wall anyway,” he later told the crowd. Beto O’Rourke, the former Texas congressman from El Paso who is considering a run for the White House in 2020, held a rally of his own less than a mile away.