(CNN) – House Budget Committee Chairman Paul Ryan thanked President Obama for hosting a "frank discussion about Washington's budget challenges" when the two met Thursday.
Obama, who dined with 12 Republican senators Wednesday night, invited Ryan and Democratic Rep. Chris Van Hollen, the ranking member on the budget committee, to the White House for lunch.
In a statement, Ryan said he believes "everyone needs to be a part" of the conversation surrounding the country's debt and deficit issues.
"We need an open debate about how best to balance the budget and expand opportunity," he continued. "I look forward to having that debate next week with specific budget proposals from House Republicans and Senate Democrats."
Ryan, the 2012 GOP vice presidential nominee, is expected to introduce specific proposals in his budget on Tuesday.
Appearing on CNN's "The Situation Room," Van Hollen said the meeting lasted more than an hour and included only the three men, without any aides at the table.
"It was a good lunch, it was a good meal and it was a very good conversation," he said. "The president is trying to open every avenue of dialogue to try and get something done."
The meeting marked the first time Obama sat down with Ryan since the two competed head on in the presidential race last fall.
"Their relationship is developing very well," Van Hollen said, without going into detail about the meeting.
A White House official said the president was "pleased" to host Ryan and Van Hollen, adding that the group had a "constructive discussion."
Obama, the official said, "expressed his desire to continue a dialogue with them in the weeks ahead."
On the menu was lentil vegetable soup and broiled sea bass served with a roasted vegetable ragu, according to the White House.
The meeting came one day after the GOP-controlled House passed a Republican measure to keep the government funded through September, while softening the impact of forced spending cuts on the military and veterans affairs program. The continuing resolution now goes to the Senate for consideration.
In his dinner with a dozen Republicans Wednesday night, the conversation also focused on the country's budget issues–and many senators walked away from the meeting describing it as a cordial and genuine exchange.
The bipartisan meetings signal a shift in approach for the president, as they come less than a week after both sides were engaged in bitter, partisan fight over the failure to produce a deficit-reduction plan that averted $85 billion in across-the-board cuts from going into place.
Obama on Friday placed blame on Republicans for not compromising, while Republicans faulted Democrats for the same reason.
Asked if the latest Republican outreach was a change in strategy, White House Press Secretary Jay Carney said on CNN's "The Situation Room" that it was "a change in circumstances."
"And a strategy that is born out of that change in circumstances," Carney continued.
Because there's no impending fiscal deadline, Carney added, then there's "a little room and time for constructive conversation."
Carney said the president was specifically reaching out to Republicans in Congress who have expressed an openness to moving forward.
"The president has been...having phone conversations and other kinds of encounters with lawmakers of both parties and we don't read all of them out and we won't read most of them out because, as the president said to me earlier with regards to the dinner, part of the effort here is to foster an environment where these conversations can be held in a way that allows for a free exchange of ideas," Carney said earlier in the day during the press briefing.
In an interview with CNN Chief Political Correspondent Candy Crowley, House Democratic Leader Nancy Pelosi applauded Thursday the president's efforts to meet with Republicans.
"He’s been very bipartisan in his approach. So, I think that these meetings are not something to say, ‘well, I’ll do this with you now, and do that with them later.’ I think it is, ‘let’s get some things done together to make elections less important.’ Let’s come together for the benefit of the American people, first and foremost, that’s our responsibility."
She continued: "So if he can diffuse some of their opposition to some of these issues, bravo, for the, again, for the American people."
Catch more from the interview with Pelosi Sunday on CNN's "State of the Union."
– CNN's Deirdre Walsh, Jim Acosta and Ashley Killough contributed to this report.