“We have to come up with something that can pass and that president will sign,” urged Sen. Lamar AlexanderAndrew (Lamar) Lamar AlexanderWeek ahead: Lawmakers near deal on children's health funding Ryan suggests room for bipartisanship on ObamaCare Time to end fiscal year foolishness MORE (R-Tenn.) who is the Republican Conference chairman. “We all know that."

Alexander noted that time to broker a compromise and push it through both houses before the Tuesday deadline had become short.

"I hope that the spirit of today and of tomorrow, and of Sunday is that we spend less time plotting about how we can defeat each other’s proposals as quickly as possible and more time working together,” said Alexander.

Sen. Kay Bailey Hutchison (R-Texas) also veered away from a hard line being drawn by some of her more conservative colleagues, noting that a split Congress would require a degree of compromise.

"Congress is split," said Hutchison. "We have a majority of Democrats in the Senate and Republicans in the House therefore, we are not going to get everything that any one of us believes is right."

The Senate is currently in period of "morning business" awaiting the outcome of the vote on the BoehnerJohn Andrew BoehnerDems face hard choice for State of the Union response Even some conservatives seem open to return to earmarks Overnight Finance: Trump, lawmakers take key step to immigration deal | Trump urges Congress to bring back earmarks | Tax law poised to create windfall for states | Trump to attend Davos | Dimon walks back bitcoin criticism MORE plan in the House.