Republican Sens. Ted CruzRafael (Ted) Edward CruzTrump can save Republicans and restore sanity to California in 2018 Cruz says Cambridge Analytica assured him its practices were legal Dem battling Cruz in Texas: ‘I can understand how people think this is crazy’ MORE and Mike LeeMichael (Mike) Shumway LeeOvernight Defense: Senate sides with Trump on military role in Yemen | Dem vets push for new war authorization on Iraq anniversary | General says time isn't 'right' for space corps Senate sides with Trump on providing Saudi military support Senate, Trump clash over Saudi Arabia 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."

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 ReidTrump presses GOP to change Senate rules Only thing Defense’s UFO probe proves is power of political favors Nevada Democrat accused of sexual harassment reconsiders retirement: report MORE (D-Nev.) and Senate Minority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellCollins: 'Extremely disappointing' ObamaCare fix left out of spending deal House poised to vote on .3T spending bill Budowsky: Stop Trump from firing Mueller 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 GrahamDems aim to turn ObamaCare hikes into election weapon Steyer brings his push to impeach Trump to town halls across the nation Trump formally sends Pompeo nomination to Senate 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 McCainZuckerberg: Maybe tech should face some regulations Schiff mocks Trump: Obama, Bush didn't need staff warning 'do not congratulate' Putin GOP senator tears into Trump for congratulating Putin 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 AyotteAudit finds US Defense Department wasted hundreds of millions of taxpayer dollars US sends A-10 squadron to Afghanistan for first time in three years No, the US did not spend million on a gas station in Afghanistan 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 CorkerNearly 70 percent say Trump is a bad role model for children: poll PPP poll: Dem leads by 5 points in Tennessee Senate race Dem Iraq War vets renew AUMF push on 15th anniversary of war 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 LewBig tech lobbying groups push Treasury to speak out on EU tax proposal Overnight Finance: Hatch announces retirement from Senate | What you can expect from new tax code | Five ways finance laws could change in 2018 | Peter Thiel bets big on bitcoin Ex-Obama Treasury secretary: Tax cuts 'leaving us broke' 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 Boehner4 reasons Mike Pompeo will succeed at Foggy Bottom The misunderstood reason Congress can’t get its job done GOP sees McCarthy moving up — if GOP loses the House MORE (R-Ohio) to bring it up for a vote.

With Tea Party conservatives already lining up in opposition, BoehnerJohn Andrew Boehner4 reasons Mike Pompeo will succeed at Foggy Bottom The misunderstood reason Congress can’t get its job done GOP sees McCarthy moving up — if GOP loses the House 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.