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The Senate Foreign Relations Committee is expected to approve the bill next Tuesday and hold a full Senate vote soon after. The measure provides a skeptical Congress with the opportunity to review the deal and prevents the president from waiving congressional sanctions on Iran during the review period.

The White House has engaged in an extensive lobbying campaign on the Hill and with constituent groups concerned about Iran ever since a framework agreement was announced on April 2. The U.S. and the other world powers negotiating with Tehran have until June 30 to hammer out the final details to seal a deal.

READ: Schumer: Let Congress decide on Iran deal

But so far, the administration hasn't been able to win over all the Senate Democrats, many of whom believe strongly that Congress has a constitutional obligation to weigh in on a major nuclear agreement with a long-time enemy of the United States.

The bill already has nine Democratic co-sponsors and a handful of other Democrats have either expressed support or remain open to backing the bill. When combined with the Senate Republicans and one independent who support the legislation, that leaves backers just four shy of the 67 needed to sustain the veto that Obama has promised.

The bill's proponents got an important boost with the recent news that New York Sen. Chuck Schumer, a co-sponsor, was in line to become the Senate's Democratic leader in 2017. The expected ascension has put his support in the spotlight since the popular Schumer could clear the way for more Democrats to back the bill.

Schumer, a staunch supporter of Israel, also represents a large Jewish constituency, many of whom have expressed concerns about the framework Iran deal.

The looming Iran agreement is vehemently opposed by Israeli Prime Minister Benjamin Netanyahu, and the GOP believes the split among Jewish voters on the issue will allow it to make inroads within the demographic group, which usually votes heavily Democratic. Schumer's bucking the White House may signal to Jewish voters that he'll be fighting to get the best Iran deal possible.

What's not clear is whether Democrats who vote for the bill, co-authored by Foreign Relations Committee Chairman Bob Corker and one of the committee's senior Democrats, Robert Menendez of New Jersey, would also vote to override the veto of a Democratic president. CNN asked Schumer last month whether he would defy the President with a vote to override, but he refused to answer.

"I'm not going to deal in hypotheticals," Schumer responded.

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Senate nears veto-proof tally on Iran bill - CNN.com