WASHINGTON—President Barack Obama on Tuesday crossed a key threshold in building support for the Iran nuclear deal, winning the backing of enough Senate Democrats that he could avoid having to use a veto to advance the accord.

The new support, of 42 senators, means a shift in focus in the debate over the agreement, which was reached in Vienna in July. What had been a struggle to win over undecided lawmakers turned Tuesday into wrangling over procedure.

Senate Majority Leader Mitch McConnell (R., Ky.) had been planning a formal floor debate over the Iran deal. But Senate Minority Leader Harry Reid (D., Nev.) now has enough support to block the legislation—technically, a resolution disapproving the deal—on procedural grounds, and he worked Tuesday to keep Democrats in line behind the strategy. Such a move would spare Mr. Obama from having to implement a significant piece of American foreign policy—and his top second-term priority—with a veto.

Republicans led by Mr. McConnell say the Senate should have a full debate over the measure, pointing to legislation enacted just months ago that gave Congress a say over the Iran deal. “We should conduct a respectful and serious debate that’s consistent with the serious ramifications of this agreement,” he said on Tuesday. Some Senate Democrats have suggested they might insist on full debate and a vote on the resolution even though they support the deal.

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