WASHINGTON — Ten Senate Democrats who have advocated putting more sanctions on Iran gave the White House a two-month reprieve on Tuesday, saying they would wait until after the late-March deadline for completing the outlines of a deal to restrain Tehran’s nuclear program before voting for a bill that President Obama has said would undermine any chance of reaching an agreement.

The concession came in a letter to Mr. Obama from Senator Robert Menendez, the ranking Democrat on the Foreign Relations Committee, who has been increasingly at odds with the White House for his strong advocacy of a bill intended to squeeze Iran’s oil revenues even harder if the nuclear negotiations do not result in an agreement. But Mr. Menendez’s letter made clear that he and the other Democrats would join Senate Republicans if a meaningful accord was not reached by March 24.

While Secretary of State John Kerry has expressed optimism that an extension agreed upon in November would lead to a speedy agreement, talks have been largely stalemated since then, and few of Mr. Kerry’s aides now believe that deadline can be met. The ultimate deadline for the talks is the end of June, by which time the two countries are supposed to develop a detailed accord.

Mr. Menendez wrote in the letter that he and the other senators “remain deeply skeptical that Iran is committed to making the concessions required to demonstrate to the world that its nuclear program is exclusively peaceful by March 24.” But he said that in deference to Mr. Obama’s fears that any new sanctions would undermine negotiations, “we will only vote for this legislation on the Senate floor if Iran fails to reach agreement on a political framework that addresses all parameters of a comprehensive agreement.”