Some GOP lawmakers said they hoped constituents would take advantage of the recurring votes to pressure Democratic senators to change their minds. “Hopefully the American people express their will to the knuckleheads who are blocking progress,” in the Senate, said Rep. Mo Brooks (R., Ala.). Many House Republicans said it is up to the Senate to make the next move and send the best bill they can to the House.

But other GOP lawmakers said Senate Republicans were confronting the limits of their power. “I know a lot of people want them to quote fight harder,” said Rep. Tom Cole (R., Okla.) “You can fight as hard as you want, you can’t turn 54 into 60.”

Democrats have said they would only pass a bill extending the agency’s funding without any polarizing immigration measures attached to it. “Republicans should not and can not risk a shutdown of a large part of the government tasked with keeping us safe just to appease the hard right of their party who oppose the president’s policies,” said Sen. Charles Schumer (D., N.Y.)

In the past, Mr. Boehner has turned to House Democrats to help pass spending bills needed to keep the government running in the face of GOP resistance. But GOP aides and lawmakers said Mr. Boehner won’t bring a bill just funding the Homeland Security to the House floor now, a move that would be seen as an embarrassing capitulation less than two months after Republicans claimed full control of Congress.