Associated Press Photo/Andrew Harnik
Secretary of State John Kerry testifies along with Secretary of Energy Ernest Moniz, and Secretary of Treasury Jack Lew at a Senate Foreign Relations Committee hearing on Capitol Hill, in Washington, Thursday, July 23, 2015, to review the Iran nuclear agreement. The hearing marked a new phase of a bruising struggle that will lead to what will arguably be the biggest foreign policy vote in more than a decade.

WASHINGTON — Supporters of the Iran nuclear deal see growing momentum on their side in the Senate, raising the possibility they'll be able to block a disapproval resolution and protect President Barack Obama from having to use his veto pen.

Such an outcome — which looked all but inconceivable in the days after the deal was signed July 14 — remains a long shot. It would be a major victory for Obama, who is staking his foreign policy legacy largely on the agreement struck by the U.S., Iran, and five world powers to dismantle most of Iran's nuclear program in exchange for billions in sanctions relief.

It would take 41 senators to block the disapproval resolution scheduled for a vote next month; only 34 lawmakers would be required to uphold an Obama veto of such a resolution.

Sen. Patty Murray, D-Wash., on Tuesday, became No. 29 on a list of Democrats and independents who have publicly announced their support of the deal.

"This is not a perfect deal, and there are several elements I would like to be stronger," Murray said. "But after working my way through the details and the alternatives, losing a lot of sleep, and having a lot of good conversations with so many people, I am convinced that moving forward with this deal is the best chance we have at a strong diplomatic solution. It puts us in a stronger position no matter what Iran chooses to do, and it keeps all of our options on the table if Iran doesn't hold up their end of the bargain."

Two Senate Democrats — New York's Chuck Schumer and New Jersey's Bob Menendez — have announced that they will vote against the agreement. But supporters feel confident that they can get to 34, and some have begun to say in private that 41 may even be within reach.

Reaching 41 remains a remote possibility, however, with Republicans unanimously opposed and Israeli officials arguing vehemently against a deal they say could empower enemies sworn to their destruction. And yet predictions that Republican opponents and the powerful-pro-Israel lobby would use Congress' August recess to make the deal politically toxic have not come to pass.

Although polls register significant public concerns about the agreement, undeclared Democratic senators have increasingly broken in favor. The latest to do so were Senate Minority Leader Harry Reid of Nevada, who announced his support over the weekend, and Sen. Debbie Stabenow of Michigan, who followed on Monday.

"We feel good about the fact that after two-thirds of the Democratic caucus has committed that we have substantial support for the president with only two dissenters," Sen. Dick Durbin, D-Ill., who's leading the whip operation in favor of the deal, said in an interview. "We continue to work it. I think the two that came out this week so far, Harry Reid and Debbie Stabenow, are very helpful," Durbin said, though he declined to predict success.

Reid told reporters in Las Vegas attending a clean energy conference Monday that he expects to see a couple more "yes" votes in the next couple of days.

"I know it's a long shot, I hope that it can be done," he said of prospects for blocking the disapproval resolution. "We'll just have to see. Because right now, it's based on a whole lot of uncounted votes."

Several key senators have yet to announce their position. One of the most-watched is Sen. Ben Cardin of Maryland, who is the top Democrat on the Foreign Relations Committee and whose position could influence colleagues.

Congressional aides say they've rarely seen the White House work an issue so hard, with Obama making personal appeals to undecided lawmakers.

"We've been leaders in nonproliferation, which is part of the reason why I am so insistent that Congress not block a historic diplomatic effort when it comes to making sure that Iran does not get a nuclear weapon," the president said at a fundraiser Monday night in Henderson, Nev.

The disapproval resolution is certain to pass the House, though Democratic leader Nancy Pelosi has declared that Democrats have the votes to sustain an Obama veto. Two-thirds votes are required in each chamber to override a presidential veto.

Jeremy Ben-Ami, president of J Street, a liberal Jewish group supporting the deal, said his organization is telling senators that since the deal clearly seems to command the support to prevail in the end, they should think twice about voting for a resolution of disapproval that would only end up getting vetoed.

"The odds are absolutely against getting to 41, and no one would have thought that possible at the beginning," Ben-Ami said. "But it's certainly not out of the realm of possibility."

A spokesman for the American Israel Public Affairs Committee, which is spending millions to try to kill the deal, maintained opponents will prevail.

"In poll after poll, the American people reject this flawed agreement and we are confident that so will a strong and growing bipartisan majority in Congress," said AIPAC spokesman Marshall Wittmann.