CNN Chief National Correspondent John King told CNN's "New Day" that Ryan, the Republican chairman of the House Budget Committee, "said something to the effect of, 'Look, we know you don't like our position, we know you probably don't respect our position, but we're the Republican majority.' "
Those comments changed the tone of the meeting, King said. Obama reiterated his opposition to negotiations before the government reopens, then urged GOP leaders to go ask their colleagues what they want done to make that happen.
Republican Rep. Steve Southerland of Florida said the interaction between the two men -- who squared off in the 2012 election, when Ryan ran as his party's vice presidential nominee -- "was an important part of the conversation."
"Paul and the president certainly have a past through the last election, and I think there's a great respect between them. And you can't make that up," Southerland said.
The new atmosphere was reflected in both sides' comments after the meeting. The White House called it good and said the parties discussed "potential paths forward." GOP Rep. Hal Rogers of Kentucky said they were talking "in good faith" about both the debt and reopening the government.
"We're all working together now," said Rep. Pete Sessions, a Texas Republican.
In a video message to a right-wing political summit on Friday, Ryan noted conservatives can't get everything they want with Democrats holding the White House and a majority of the Senate.
"This president won't agree to everything we need to do," said the message from Ryan, according to excerpts provided by one of his aides. "A budget agreement with this president and this Senate won't solve our problems. But I hope it's a start."