Washington (CNN) Over the past two weeks, President Barack Obama made a full-court press to persuade lawmakers to back his Iran diplomacy and abandon a bill giving Congress a say in any final deal.

Ultimately, though, the White House was forced to bow to the reality: Enough Democrats supported the principle of allowing Congress to weigh in on the nuclear deal that even a veto threat couldn't stop it.

Shortly before the Senate Foreign Relations Committee unanimously voted Tuesday to send the bill to the Senate floor, the White House indicated that the veto threat had been withdrawn.

Republicans and some Democrats chalked up the about-face to a calculation by the administration that the numbers simply weren't on its side.

"I don't think they had a choice. The votes were there," said Arizona Republican Sen. Jeff Flake, a committee member. "I think they certainly didn't want to flirt with the situation where there would be enough votes to override a veto. That's where we were headed."

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As California Sen. Dianne Feinstein, the top Democrat on the Senate intelligence committee, put it: "I think it was a recognition of the lay of the ground, so to speak."

But the White House is pushing back on the suggestion that it caved on the bill.

"Despite the things about it that we don't like, enough substantial changes have been made that the President would be willing to sign it," White House Press Secretary Josh Earnest said Tuesday, "because it would reflect the kind of compromise that he'd be willing to sign."

According to the administration, those changes include shortening a 60-day review period to a maximum of 52 days, dropping the requirement that Iran declare it will not sponsor terrorism and other similar poison-pill language, and wiping out "other extraneous elements" from the measure.

'They did not cave'

"They did not cave. They did not cave," said Maryland Sen. Ben Cardin, the top Democrat on the Senate Foreign Relations Committee. "We looked at the language and got the language to where they were comfortable."

Despite the White House's massive outreach across Capitol Hill -- the administration placed more than 130 calls, many from President Barack Obama, Vice President Joe Biden and Secretary of State John Kerry, while several Cabinet secretaries held briefings for lawmakers -- it was the close discussions with Cardin that resolved many of the White House's concerns.

Cardin was the key point person in hammering out the compromise version with Senate Foreign Relations Committee Chairman and lead bill sponsor Bob Corker of Tennessee.

While Cardin said the administration got most if not all of what it was looking for, the Republican senator maintained that the legislation remained substantially the same as what he had originally drafted.

"The bill is slightly different but we're talking about a gnat's hair difference," Corker said. "It gave them (the White House) a chance to say changes had been made even though the essence is what it's always been."

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That essence includes requiring the details of any Iran deal -- due by the end of June -- be sent to Congress and paving the way for a vote on a joint resolution that could express approval or disapproval of the deal. During Congress's review, the President wouldn't have the ability to waive congressionally mandated sanctions on Iran.

While originally the review period was set to last 60 days, the compromise dials that down to 30 days, with 12 more days added automatically if Congress passes a bill and sends it to the President, and an additional 10 days of congressional review if Obama vetoes that legislation.

If the deal is submitted late, after July 9, the review period reverts to 60 days.

Obama must certify Iran's compliance

Either way, Obama is required to certify to Congress every 90 days that Iran is complying with the terms of the deal.

The bill also requires the President to make a series of detailed reports to Congress on issues ranging from Iran's nuclear program to its ballistic missile work and its support for terrorism globally, particularly against the U.S. and its allies.

When it came to getting the White House to drop its veto threat, Republican Sen. Ron Johnson of Wisconsin acknowledged that the math also aided Obama: He only needs 34 senators to agree with him to sustain a veto of the bill, which would be read as "basically tacit approval of the deal."

"That's a pretty low threshold," he said. "He probably came to the conclusion that no matter what the deal is, he can hurdle that bar."

Massachusetts Sen. Ed Markey, another committee Democrat, also thought that number helped sway the administration.

Still, he said, "I'm sure they were not thrilled with ultimately having to cede this."

Markey said the changes to the bill that Cardin was able to work out with Corker also helped many Democrats who shared the administration's concerns stay on board.

"All of the concerns that members had been raising on the Democratic side were now gone, and so as a result the argument came down to whether there would be a congressional role at all," he explained. "It made it a lot easier for those Democrats who had been raising objections on terrorism certification and other issues to then feel more comfortable voting yes."

That in turn gave the bill too much Democratic support for the president to be secure in a veto.

Virginia Democratic Sen. Tim Kaine, a member of the Foreign Relations Committee who has been pressing to give Congress the chance to vote on a deal, said it became increasingly clear in his recent discussions with the administration that the main issue wasn't whether Congress would be involved, but how to structure a process both sides felt was workable.

"It's really a choice between congressional involvement under a prompt and reasonable set of rules or a free-for-all involvement," he said. "I think the events of the last couple of months convinced them -- yep, it might not have been our original choice, but we'd rather have it circumscribed and appropriate."