Republican Sens. Ted CruzRafael (Ted) Edward CruzMatthew McConaughey on potential political run: 'I'm measuring it' Professor tells Cruz that Texas's voter ID law is racist Schumer moves to break GOP blockade on Biden's State picks MORE and Mike LeeMichael (Mike) Shumway LeeThe Hill's 12:30 Report - Presented by Facebook - Dems attempt to tie government funding, Ida relief to debt limit Trump lawyer offered six-point plan for Pence to overturn election: book Graham found Trump election fraud arguments suitable for 'third grade': Woodward book 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 Mason ReidTo Build Back Better, we need a tax system where everyone pays their fair share Democrats say Biden must get more involved in budget fight Biden looks to climate to sell economic agenda MORE (D-Nev.) and Senate Minority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellOn The Money — Democrats rush to finish off infrastructure Biden employs flurry of meetings to unite warring factions GOP senators say Biden COVID-19 strategy has 'exacerbated vaccine hesitancy' 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 GrahamThe Hill's 12:30 Report - Presented by Facebook - Biden meets with lawmakers amid domestic agenda panic The Hill's Morning Report - Presented by Alibaba - House Democrats plagued by Biden agenda troubles Graham tries to help Trump and McConnell bury the hatchet 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 McCainCollins to endorse LePage in Maine governor comeback bid Meghan McCain: Country has not 'healed' from Trump under Biden Biden steps onto global stage with high-stakes UN speech 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 AyottePoll: Potential Sununu-Hassan matchup in N.H. a dead heat  Democrats facing tough reelections back bipartisan infrastructure deal Sununu seen as top recruit in GOP bid to reclaim Senate 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 CorkerCheney set to be face of anti-Trump GOP How leaving Afghanistan cancels our post-9/11 use of force The unflappable Liz Cheney: Why Trump Republicans have struggled to crush her  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 LewThe Hill's Morning Report - Biden argues for legislative patience, urgent action amid crisis On The Money: Senate confirms Yellen as first female Treasury secretary | Biden says he's open to tighter income limits for stimulus checks | Administration will look to expedite getting Tubman on bill Sorry Mr. Jackson, Tubman on the is real 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 BoehnerLobbying world A new kind of hero? Last week's emotional TV may be a sign GOP up in arms over Cheney, Kinzinger MORE (R-Ohio) to bring it up for a vote.

With Tea Party conservatives already lining up in opposition, BoehnerJohn Andrew BoehnerLobbying world A new kind of hero? Last week's emotional TV may be a sign GOP up in arms over Cheney, Kinzinger 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.