Republican Sens. Ted CruzRafael (Ted) Edward CruzDebbie Wasserman Schultz marks 10 years as breast cancer survivor Foreign agent registration is no magical shield against Russian propaganda Let Trump be Trump and he'll sail through 2020 MORE and Mike LeeMichael (Mike) Shumway LeeSupreme Court takes on same-sex wedding cake case House approves motion to go to tax conference — with drama Trump really will shrink government, starting with national monuments 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 ReidBill O'Reilly: Politics helped kill Kate Steinle, Zarate just pulled the trigger Tax reform is nightmare Déjà vu for Puerto Rico Ex-Obama and Reid staffers: McConnell would pretend to be busy to avoid meeting with Obama MORE (D-Nev.) and Senate Minority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellGOP strategist donates to Alabama Democrat McConnell names Senate GOP tax conferees Brent Budowsky: A plea to Alabama voters 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 GrahamGOP and Dems bitterly divided by immigration We are running out of time to protect Dreamers US trade deficit rises on record imports from China 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 McCainGOP strategist donates to Alabama Democrat Meghan McCain knocks Bannon: 'Who the hell are you' to criticize Romney? Dems demand Tillerson end State hiring freeze, consult with Congress 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 AyotteExplaining Democratic victories: It’s gun violence, stupid Trump voter fraud panel member fights back against critics Dems plan to make gun control an issue in Nevada 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 CorkerFormer Dem Tenn. gov to launch Senate bid: report McConnell 'almost certain' GOP will pass tax reform Former New Mexico gov: Trump's foreign policy is getting 'criticized by everybody' 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 LewSenator demands answers from DOJ on Russia bribery probe Koskinen's role in the ObamaCare bailout another reason Trump must terminate him The debt limit is the nation's appendix — get rid of it 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 BoehnerJohn Feehery: A political forest fire Trump's pick for Federal Reserve chief is right choice at right time The two-party system is dying — let’s put it out of its misery MORE (R-Ohio) to bring it up for a vote.

With Tea Party conservatives already lining up in opposition, BoehnerJohn Andrew BoehnerJohn Feehery: A political forest fire Trump's pick for Federal Reserve chief is right choice at right time The two-party system is dying — let’s put it out of its misery 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.