Republican Sens. Ted CruzRafael (Ted) Edward CruzSenate Dems hold floor talk-a-thon against latest ObamaCare repeal bill Overnight Finance: CBO to release limited analysis of ObamaCare repeal bill | DOJ investigates Equifax stock sales | House weighs tougher rules for banks dealing with North Korea GOP state lawmakers meet to plan possible constitutional convention MORE and Mike LeeMichael (Mike) Shumway LeeOvernight Defense: Senate passes 0B defense bill | 3,000 US troops heading to Afghanistan | Two more Navy officials fired over ship collisions Senate passes 0B defense bill Overnight Finance: CBO to release limited analysis of ObamaCare repeal bill | DOJ investigates Equifax stock sales | House weighs tougher rules for banks dealing with North Korea MORE said Wednesday that they won't try to block the bipartisan deal to reopen the government and raise the debt ceiling.

"The timing of the vote will make no difference to the outcome," Cruz (Texas) said. "I have no intention of delaying the vote."

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Cruz and Lee (Utah) had spearheaded the campaign to attach ObamaCare language to any new budget bills, and had pushed their colleagues to take a hard-line stance. 

But the pair emerged from a meeting of Senate Republicans Wednesday afternoon to say they would let the budget agreement — which contains only one provision on ObamaCare — move forward.

"Some of us want to see it first — we still don't have text," Lee said. "But provided we get the text and have time to review it and everything, I suspect we'll collapse the time."

There has been speculation that the bipartisan deal might originate in the House, thereby eliminating some procedural hurdles when the bill moved to the Senate. But with Cruz and Lee withdrawing their filibuster threats, several GOP senators said the upper chamber will likely act first on the agreement.

The deal reached by Senate Majority Leader Harry ReidHarry ReidThe Memo: Trump pulls off a stone-cold stunner The Memo: Ending DACA a risky move for Trump Manchin pressed from both sides in reelection fight MORE (D-Nev.) and Senate Minority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellSenate passes 0B defense bill Overnight Health Care: New GOP ObamaCare repeal bill gains momentum Overnight Finance: CBO to release limited analysis of ObamaCare repeal bill | DOJ investigates Equifax stock sales | House weighs tougher rules for banks dealing with North Korea MORE (R-Ky.) would fund the government through Jan. 15 and raise the debt ceiling until February. It would require income verification for subsidies under ObamaCare, but otherwise leave the healthcare law untouched.

The weeks-long fight over ObamaCare has left the GOP sharply divided and licking its wounds, after centrist Republicans warned for weeks that the battle was futile and damaging to the party.

Several Republicans left the meeting Wednesday grumbling that the conservatives in their ranks had prolonged the impasse even as Republicans were sinking in the polls.

"We left a lot on the table because we couldn't get our act together, but this is the best Mitch could do," said Sen. Lindsey GrahamLindsey Olin GrahamTop Louisiana health official rips Cassidy over ObamaCare repeal bill Senate Dems hold floor talk-a-thon against latest ObamaCare repeal bill Overnight Defense: Senate passes 0B defense bill | 3,000 US troops heading to Afghanistan | Two more Navy officials fired over ship collisions MORE (R-S.C.), who will vote for the deal. "We need to stop the bleeding, lessen the damage to the party."

Sen. John McCainJohn Sidney McCainSenate's defense authorization would set cyber doctrine Senate Dems hold floor talk-a-thon against latest ObamaCare repeal bill Overnight Defense: Senate passes 0B defense bill | 3,000 US troops heading to Afghanistan | Two more Navy officials fired over ship collisions MORE (R-Ariz.) echoed that message.

"I hope it is another 15 years at least before we have to go through this exercise again," McCain said. "We're in a hole, we have to dig out and come up with a positive agenda."

A number of Republicans have emerged in recent days to pressure Cruz to back down from his insistence that ObamaCare be scaled back considerably as part of a budget deal.

"I would hope that whatever comes forward, that we would allow a vote on it as soon as possible," Sen. Kelly AyotteKelly Ann AyotteStale, misguided, divisive: minimum wage can't win elections Trump voter fraud commission sets first meeting outside DC RNC chair warns: Republicans who refused to back Trump offer 'cautionary tale' MORE (R-N.H.) told CNN Wednesday morning.

Cruz, for his part, denied that he ever aimed to block the vote.

"I never had any intention to delay this vote," he said. "I thought it was interesting to read in the papers speculation that others were asking me not to delay this vote because no one in Republican leadership ever asked."

Sen. Bob CorkerRobert (Bob) Phillips CorkerOvernight Defense: Senate passes 0B defense bill | 3,000 US troops heading to Afghanistan | Two more Navy officials fired over ship collisions Senate passes 0B defense bill Corker pressed as reelection challenges mount MORE (R-Tenn.) said he never thought Cruz and others would filibuster the McConnell deal.

"At some point when you've reached the end of the road to then make life miserable for no reason ... then the pendulum swings," Corker said.

Corker said he would not "tout" the deal as a GOP victory but said that it at least preserved the sequester in the Budget Control Act, which he said was perhaps the best thing he had ever voted for.

"I am not going to champion it, all I am going to say is we did not go backwards," he said.

Wednesday's breakthrough follows weeks of high-stakes wrangling between the GOP-controlled House and Democratically controlled Senate over how to fund the government and hike the $16.7 trillion debt ceiling. Treasury Secretary Jack LewJacob (Jack) Joseph LewBipartisan bill would force Treasury to put Tubman on bill Top conservative rails against ‘clean’ debt limit increase Trump mocked Obama for three chiefs of staff in three years MORE has warned that the government would be unable to pay all of its bills beyond Wednesday if Congress didn't act first.

If the Senate passes the package on Wednesday, as expected, the bill would move to the House, all but forcing Speaker John BoehnerJohn Andrew BoehnerSpeculation mounts, but Ryan’s job seen as safe Boehner warns Trump: Don't pull out of Korea-US trade deal GOP Rep: Ryan wasting taxpayers dollars by blocking war authorization debate MORE (R-Ohio) to bring it up for a vote.

With Tea Party conservatives already lining up in opposition, John BoehnerJohn Andrew BoehnerSpeculation mounts, but Ryan’s job seen as safe Boehner warns Trump: Don't pull out of Korea-US trade deal GOP Rep: Ryan wasting taxpayers dollars by blocking war authorization debate MORE will likely need an overwhelming number of House Democrats to get the package to President Obama.

On the walk from his office to the House chamber on Wednesday at noon, Boehner was silent in the face of questions about the emerging deal..

Erik Wasson, Peter Schroeder, Alex Bolton, Elise Viebeck and Sam Baker contributed.

— This story was last updated at 1:36 p.m.