Dreier's comments are among the first from any Republican implying that the next step is to work in the traditional, bi-cameral way with the Senate on a compromise debt ceiling solution. It's also the latest indication that the House and Senate will spend the weekend on a compromise, which could make for a busy Monday and Tuesday trying to get the compromise back through the House and Senate.

BoehnerJohn Andrew BoehnerHillicon Valley: Trump hits China with massive tech tariffs | Facebook meets with GOP leaders over bias allegations | Judge sends Manafort to jail ahead of trial | AT&T completes Time Warner purchase Facebook execs to meet with GOP leaders over concerns about anti-conservative bias Boehner: Federal government should not interfere in recreational marijuana decisions MORE's amended bill would increase the debt ceiling by $900 billion right away, and allow for another $1.6 trillion increase upon congressional passage of a balanced budget amendment to the Constitution.

But Democrats, during one of the last stages of House debate on the Boehner plan, argued that House passage of a bill that Senate Democrats oppose only adds to the risk of a government default.

Rep. Louise Slaughter (D-NY) said Senate Democrats and the White House have said "repeatedly" that they oppose the bill, and that Republicans are playing a dangerous game by pushing for their bill with less than five days left before the August 2 debt ceiling deadline.

"They have said it repeatedly, they have said it clearly," Slaughter said. "Yet the majority continues to believe this bill can actually avert the danger of default. They're playing a dangerous game of chicken, asking the nation to give into their demands if we want the American economy to live to see another day."

Slaugther also discussed the balanced budget amendment requirement as a hostage situation that Democrats should not accept.

"In effect, this legislation releases one hostage and takes another," she said. "Six months from now, we will be forced to choose between a constitutional amendment and putting the nation back on the brink of default. I refuse dot trade hostages with the majority and prolong this crisis for another six months."