But that Democratic support likely comes with a cost, members said. Many Democrats are demanding that the measure be amended so it doesn’t kill the deal before it can be finalized by a June 30 deadline. So the onus is on Republicans to work with Democrats — particularly if they want to assemble a 67-vote veto-proof majority — although it’s not clear exactly what legislative changes would preserve the complex and still-evolving agreement.

Independent Maine Sen. Angus King, who caucuses with Democrats and is a cosponsor of the bill, offered a glimpse inside the caucus’s thinking. Asked if he would still vote for it, King replied: “‘Yes, but …’ is my answer.”

“It depends how it’s handled the next two to three weeks,” King said in an interview on Friday. “I’m not in if it’s a partisan weapon.”

Nonetheless, Senate Foreign Relations Chairman Bob Corker (R-Tenn.) said Friday he’s not backing away from an April 13 markup of his legislation that would allow Congress to review the agreement for 60 days and potentially reject the lifting of sanctions on Tehran, which would effectively kill the deal. Corker said he’s open to changing the bill, although he’s going to have to balance Democratic demands with hawks in his own party who would prefer to scuttle the agreement.

It may be late April or even May before Senate Majority Leader Mitch McConnell (R-Ky.) can bring the Iran bill to the floor. First, the Senate has to deal with a Medicare payments bill, resolve an impasse over a stalled human trafficking bill and confirm a new attorney general when the chamber returns in mid-April.

That means administration officials and the president himself will have time to work with their Democratic allies and try to slow the bill down or at least limit its scope so it doesn’t interfere with drafting a final nuclear deal by the end of June.

“The White House has decided that they can’t avoid congressional review altogether,” said one senior Democratic aide. “Democrats want to talk about what tweaks they can make that the administration can live with.”

Still, the official line from the White House is that the president would veto the bill. “Our position on that has not changed,” said press secretary Josh Earnest. “We believe that this is clearly the purview of the president.”

The longer the two parties haggle, though, the more time Republicans have to criticize the president, the deal and reluctant Democrats, who don’t want the framework to fall apart because of their votes.

According to Corker, the current bill wouldn’t upend the deal. “They want to make sure our bill completely aligns with the way the political framework is. But it does. Every day that goes by I feel better about the substance of legislation,” he said in an interview.

The battle will play out in Corker’s committee in mid-April. He’s open to voting on a number of amendments, but changes viewed as too administration-friendly or controversial could lead to a stalemate, since there are Republicans on the committee who want to move the measure in the opposite direction.

Earlier versions of Corker’s bill essentially gave Congress veto authority over the deal. But the Tennessee Republican has softened the measure so it allows Capitol Hill to vote only on whether to lift sanctions and requires the White House to file quarterly reports on Iran’s adherence to a deal.

Democrats sound like they want more changes, although they haven’t yet described what those modifications would be.

“Congress should have a vote, should weigh in. I’m looking for the administration to reach out and fully inform members of the committee,” said Sen. Chris Coons (D-Del.), who spoke about Iran with the president last month and with Deputy Secretary of State Tony Blinken on Thursday evening. “I do not want to undermine the prospects for a strong, enforceable verifiable deal.”

A member of Corker’s committee, Coons has signaled he could support Corker’s bill, but he’s not officially on board yet. His support, he said Friday, “depends on the changes made to the bill, and it depends on the level of engagement on the White House.”

Republicans worry that Democrats are looking for excuses to side with the administration rather than take a confrontational stance toward Iran.

“So many are voices of support of Israel. But I think in too many cases it’s lip service,” said Sen. Ron Johnson (R-Wis.).

Despite the partisan rhetoric, Corker is confident the bitterly divided Senate can come together behind his bill. “Certainly there’s been some bellicose comments, certainly there’s been some on both sides,” Corker said. “But I think that by and large those who are watching understand how significant this is. The Senate as a whole will look at what’s before us and respond in a sober way.”

But there’s already been enough partisan squabbling on Iran to fill a book, and a shifting group of Democrats have declared common cause with Republicans before, only to be wooed away by White House lobbying.

Republicans first tried moving forward with a measure that would impose harsh sanctions on Tehran. But, sensing the direct sanctions approach was losing support with Democrats, the GOP switched tactics and produced Corker’s bill, which allows Congress to play a role in finalizing any deal.

Work on Corker’s bill has already been delayed after House Speaker John Boehner (R-Ohio) invited Israeli Prime Minister Benjamin Netanyahu to address Congress on Iran. Many Democrats viewed the speech as an insult to the president.

It wasn’t till several weeks after the speech that King, Corker and a bipartisan group of colleagues introduced the congressional approval bill.

Their bill promptly ran into a veto threat and more Democratic resistance after McConnell tried to speed the legislation to the Senate floor. After lobbying from the administration, lawmakers agreed to give President Barack Obama more time to negotiate until Corker’s April 14 committee vote.

Given the back-and-forth, perhaps it’s not a coincidence that the Iran deal was announced during the doldrums of the congressional recess. Republicans could blast the deal rhetorically but, without Congress in session, could take no official action against the White House.

Now the White House will renew its lobbying push over the next 10 days and gauge where the votes are for Corker’s bill.

“I don’t think you can say that the White House isn’t all hands on deck when it comes to Iran,” said Sen. Chris Murphy (D-Conn.). “They’ve been pretty consistently all over Democrats and Republicans.”

Indeed, Blinken has called several senators to brief them on the deal one-on-one, U.S. Ambassador to the United Nations Samantha Power has been making phone calls to senators, and Obama is reaching out to House and Senate leaders.

Corker said he’d spoken to Energy Secretary Ernest Moniz on Friday and Blinken on Thursday but hadn’t spoken to Obama since the announcement.

“I have peers telling me that the administration is weighing in, whether it’s Samantha, whether it’s [Vice President Joe] Biden, whether it’s the president himself,” Corker said. “They will work to weaken our role, but that’s part of it. I understand it.”

Sarah Wheaton contributed to this report.