“If he lets the Freedom Caucus be the tail that wags the dog, there’s no way we’ll reach an agreement that can pass the Senate,” Mr. Schumer said on the Senate floor.

Lacking the votes to approve the temporary spending measure paired with long-term military funding, the Senate could strip out the military portion and send the measure back to the House.

“I think everyone understands that this will probably end up being a Ping-Pong situation, and we’ll see where the ball lands,” said Representative Carlos Curbelo, Republican of Florida.

But Mr. Meadows was eager for a fight over the military funding. “That’s going to be up to Democrats to decide whether they want to continue to keep our military men and women hostage,” he said.

To bring an end to last month’s brief shutdown, lawmakers approved a temporary spending bill that keeps the government open through the end of Thursday. Now another stopgap measure is needed. Adding to the tight timeline facing lawmakers, House Democrats are scheduled to hold a retreat in Maryland beginning Wednesday.

“Here we are again,” Senator John Cornyn of Texas, the No. 2 Senate Republican, said on the Senate floor on Monday, before complaining about the use of one stopgap bill after another. “Governing is not a merry-go-round,” he said.

The stopgap bills have been needed because congressional leaders have yet to reach a deal to raise the caps on military and domestic spending, an impasse that has kept them from negotiating a long-term spending bill. There were signs of hope on Monday that an agreement could be within reach.