As the Obama administration faced scepticism in closed-door meetings with House and Senate members, a compromise emerged on a Senate measure that would shorten the period that Congress would have to review the final deal.
Senate Foreign Relations Committee chairman Bob Corker said the total review period would be reduced to 52 days, from the 60 days in earlier drafts of the bill.
The period includes a 30-day review, plus 12 days for a possible veto by President Barack Obama and a 10-day window for an attempt to override such a veto.
The idea is that the compromise legislation would bring more Democrats on board, potentially building a veto-proof majority.
Senate Democrat Chris Coons said he was encouraged by the shift from a 60-day review to 30 days, "so that there's less of a risk that it will drag on."
But he warned that including poison-pill amendments in the legislation - such as one being considered that would require Iran to recognise the state of Israel - would lose his support.
The framework, reached in Lausanne, Switzerland, marked a major breakthrough in a 12-year standoff between Iran and the West, which disputes Tehran's denial that it is seeking to acquire nuclear bomb.
Global powers must resolve a series of difficult technical issues by a June 30 deadline for a final deal, including the steps for lifting global sanctions imposed on Iran, and remaining questions over the possible military dimensions of its nuclear program.
Kerry made "an irrefutably compelling case on why we need to do this, and very clear how detailed this agreement is," Schakowsky told reporters after emerging from the briefing.
"I think that there is no doubt that the House of Representatives can sustain a veto if something comes from the Senate that would undermine this agreement.
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