Senators voted 18 to 4 to advance the so-called Kirk-Menendez bill, which is steadfastly opposed by the White House.
"It is clear that further action is necessary to compel Iran to reach an acceptable agreement, which is why I strongly support this critical bill," committee chairman Senator Richard Shelby said after the vote.
President Barack Obama's administration is hostile to any congressional action that might scupper the ongoing landmark negotiations between international powers -- the United States, Russia, China, Britain, France and Germany -- and Iran.
Some US lawmakers insist on keeping Iranian leaders' feet to the fire by pressing ahead with a deferred sanctions bill before June 30, which would trigger a gradual escalation of economic sanctions beginning in July should talks collapse or if Tehran violates its obligations.
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But in a victory for Obama, senators on Tuesday agreed to delay a full Senate vote until at least March 24 to allow international negotiators to reach a political framework agreement without congressional pressure.
Currently, 35 of the Senate's 100 members officially back the bill, a figure that is expected to increase in coming months.
On Tuesday, ambassadors from France, Britain, Germany and the European Union met in Washington with Senate Democrats urging them to ease the pressure.
"Let's give time to the nego(tiations)," French ambassador Gerard Araud posted on Twitter, recalling that while the March 24 date was a "goal," the true deadline for an agreement remained June 30.