Senate Foreign Relations Committee Chairman Senator Bob Corker, who wrote the bill, said the White House had agreed to go along with the bill only after it was clear there was strong Democratic support. The legislation was passed unanimously by the committee and is expected to pass the full Senate and then the House of Representatives.
"That change occurred only when they saw how many senators were going to vote for this," Corker said.
Bipartisan support for the bill had grown in recent weeks to near the 67 votes needed to override any presidential veto. But senators from Obama's Democratic Party did succeed in adding amendments to water down the bill, making it more palatable to the White House.
The bill that passed cut to 30 days from 60 the time in which Congress can review any final nuclear agreement and eliminated the requirement that Obama certify that Iran is not supporting acts of terrorism against the United States.
Instead, it requires the administration to send Congress regular, detailed reports on a range of issues including Iran's support for terrorism, ballistic missiles and nuclear program.
It requires the Obama administration to send the text of a final agreement to Congress as soon as it is completed, and blocks Obama's ability to waive many U.S. sanctions on Iran while Congress reviews the deal. It allows a final vote on whether to lift sanctions imposed by Congress in exchange for Iran dismantling its nuclear capabilities.
White House spokesman Josh Earnest said Obama, who had said in recent days he would veto the proposed bill, could accept the compromises that drew bipartisan Senate support.
"What we have made clear to Democrats and Republicans in the Senate Foreign Relations Committee is that the president would be willing to sign the proposed compromise that is working its way through the committee," Earnest said.
United Nations sanctions and those imposed by other countries could still be lifted if an agreement is reached on restricting Iran's nuclear program by a June 30 target date.
But the prospect of Congress delaying sanctions relief or rejecting it altogether would add to concerns in Iran, whose Supreme Leader Ayatollah Ali Khamenei has demanded that sanctions be lifted immediately after a deal is signed.
European officials have said that worries about a possible spoiler role for Congress have undermined the U.S. delegation's credibility in the talks and raised questions about the Obama's administration's ability to deliver on commitments.