“While this is not the agreement I would have accepted at the negotiating table, it is better than no deal at all,” Mr. Blumenthal said in a statement on Tuesday.

Sens. Ron Wyden (D., Ore.), Richard Blumenthal (D., Conn.) and Gary Peters (D., Mich.) said they would back the agreement, bringing to 41 the number of lawmakers—all members of the Democratic caucus—who endorse the accord between Iran and six other nations. That level of support could potentially block Senate Republicans from clearing the procedural hurdle necessary to send a disapproval resolution to the president’s desk, meaning that Mr. Obama might not need to use his veto to protect the deal.

“One of the reasons is that I’m confident we’re going to get 60 votes against the agreement,” said former Sen. Joe Lieberman of Connecticut, an independent who allied with Democrats and now a forceful opponent of the deal, “is because Chuck Schumer is out there against it,” referring to the New York senator and presumed next Democratic leader.

Any Democrats waiting to see which way Cardin goes will have to sit tight: He’s not expected to announce a position until late next week at the earliest, and Corker says his friend and co-leader of the committee is truly undecided.

He’s also not settling for a mere press release to make news, instead holding a public announcement on Tuesday at the University of Delaware.

In spite of the U.S. negotiating team’s hard work, this is not the best or only deal we can get. If Congress truly wants to avoid a veto, it must offer constructive changes to help make this accord worthy of our nation’s united support.

Majority Leader Mitch McConnell said the United States Congress would likely not muster the two-thirds majority needed to reject the recent accord over Iran’s nuclear program

To sustain his veto, and the agreement, Obama would need 34 Senate Democrats to vote in favor of the deal. So far, 20 have said they will do so.

Through verbal presentations regarding possible military dimensions, many in Congress are aware of the unorthodox arrangements agreed to by the International Atomic Energy Agency, the administration and our negotiating partners to keep from upsetting Ayatollah Ali Khamenei. Those actual agreements remain secret, but we know that at best they are most unusual and speak to the P5+1’s low commitment to holding Iran’s feet to the fire.

Perhaps a larger issue is beyond the scope of the deal itself. Absent a clearly articulated policy for the region, this deal will become the linchpin of the United States’ Middle East strategy. We will be relying on Iran to help achieve our goals in Iraq, Syria and perhaps elsewhere. This abrupt rebalancing could have the effect of driving others in the region to take greater risks, leading to greater instability.

Since negotiations began in earnest after President Hassan Rouhani was elected in June 2013, Iran has co-opted the Iraqi security sector, doubled down on its support of Syrian dictator Bashar al-Assad and cemented Hezbollah as an expeditionary shock force.

The personal appeals from the president are part of an orchestrated lobbying effort by the White House, supported by a coalition of antiwar and progressive organizations, aimed at persuading Democrats who are undecided about the nuclear agreement with Iran to vote against a Republican bid to block it.

Meeting weekly in a conference room in the downtown Washington office of the Ploughshares Fund, a foundation devoted to eradicating nuclear weapons, the coalition plots strategy and plans advertising, letters and petition drives to press its case. Ploughshares also finances many of the participants in the effort.

Many of them have received funding from Ploughshares, which says it has spent $11 million over the past six years advocating for a nuclear deal with Iran.

Mr. Schumer said some Democrats are discussing ways to pressure the administration to somehow toughen the deal on its own, but he said he is not involved. He also stressed that while he has made courtesy calls to some 30 senators on his decision, he is not trying to influence their vote.

As soon as Mr. Samore left, the group announced a new standard-bearer with a decidedly different message: Joseph I. Lieberman, the former senator from Connecticut and the new chairman of the group.

A congressional vote of disapproval would not necessarily be a deal breaker; in fact, it could give the administration time to improve the agreement or implement other policy measures that more effectively secure U.S. interests.

A vote of disapproval is both more and less than meets the eye. It is, on the one hand, the only way the American system allows for elected representatives to express opposition to the agreement and compel the administration to take those views into account. On the other hand, it would not kill the Iran deal unless the other parties to the agreement wanted it dead.

While a vote of disapproval would restrict the president's authority to fulfill one U.S. obligation under the accord -- waiving sanctions -- this most likely would not become a live issue until early-to-mid-2016. Until then, much could happen to change the situation, ranging from improvements in the deal that merit subsequent congressional support to new revelations of secret Iranian nuclear activity that would validate congressional skepticism.

Asked whether the United States or Iran got the better end of the agreement, 41 percent responded that the regime in Tehran gained more, while just 14 percent said the U.S. benefited more than their Iranian counterparts.

Just 27 percent say lawmakers should move to approve the deal, while 32 percent say they should vote against it.

After careful study, B’nai B’rith International has concluded that the Iran nuclear agreement is unlikely to prevent Iran from acquiring nuclear weapons. We therefore call on Congress to reject the Joint Comprehensive Plan of Action (JCPOA). As we noted when the deal was announced on July 14, it is impossible to look at Iran’s track record and trust the regime to adhere to the terms of this pact.