Dropping the requirement that the amendment pass both houses before the debt ceiling can be raised probably will cause a number of Republicans in the House to vote no. As a result, Boehner will need support from Democrats to pass the bill to make up for GOP defections.
"I think it may need anywhere from 60 to 100 Democrats," Ross said.
House Minority Leader Nancy Pelosi (D-San Francisco) issued a carefully hedged statement on that point, saying, "We all agree that our nation cannot default on our obligations and that we must honor our nation's commitments to our seniors, and our men and women in the military.
"I look forward to reviewing the legislation with my caucus to see what level of support we can provide."
In his conference call with Republican members of the House, Boehner described the deal as a considerable victory, according to sources who listened to the call. The sources spoke on the condition of anonymity to describe the conversation.
Boehner reminded the group that the president had started the debate calling for a debt limit increase with no conditions.
"Now listen, this isn't the greatest deal in the world," Boehner said, according to a source familiar with the conversation. "But it shows how much we've changed the terms of the debate in this town."
At least some of the lawmakers liked what they heard. Rep. Mike Kelly (R-Pa.) told the speaker during the conference call, "You deserve to go have a smoke."
Depending on how quickly Congress acts on the deal, a stopgap measure may be needed to allow the government to continue paying all of its bills for the next few days. If opponents of the deal decided to stage a filibuster in the Senate, that could prevent final action on the overall compromise before Tuesday's deadline.
White House senior advisor David Plouffe said earlier in the day that Obama remained open to a one- or two-day extension of the debt ceiling if Congress needed time to "dot the I's and cross the T's."
Peter Nicholas and Michael Memoli in the Washington bureau and Times staff writers Walter Hamilton and Tom Petruno in Los Angeles contributed to this report.