WASHINGTON -- After getting word in December that she’d be heading the Senate Budget Committee in 2013, Democrat Patty Murray picked up the check when she had breakfast with Republican Rep. Paul Ryan in the Senate Dining Room.

They’re nearly a generation apart with little in common, but Murray wanted to meet privately to size up the man who would soon become her latest adversary on Capitol Hill. The 62-year-old Murray, who wants to raise taxes on the wealthy and who always ranks as one of the top Senate liberals, had never met the 43-year-old Ryan, the up-and-coming chairman of the House Budget Committee from Wisconsin, who shot to the top of the Republican ranks and the No.2 spot on the 2012 GOP presidential ticket with his aggressive calls for spending cuts and no new taxes.

“I thought it was important to find out who he was and get to know him personally,” Murray recalled. “At the end of the day, we’re going to have to come together and figure out a path forward.”

Murray led the congressional supercommittee that failed to produce a deficit-reduction plan in 2011, setting the stage for the automatic spending cuts that took effect March 1. But she’s upbeat that her new starring role in the never-ending budget quagmire will produce a different result, even ending the across-the-board cuts caused by sequestration before they hit too hard.

Much will depend on her budding relationship with Ryan, even though the two are busy trading barbs, at least for now. Just Thursday, she labeled his budget a set of “failed priorities.” He said hers “cling(s) to the status quo.”

Ultimately, Murray said, one of her biggest tasks will be to ease the partisan rhetoric that has marked the budget fights, “the challenge of calming that down,” as she put it.

Negotiations between the two could begin in earnest in April, as soon as Congress returns from its traditional Easter break. The Republican-controlled House of Representatives passed Ryan’s budget plan on Thursday without a single Democrat in support. The Democratic-run Senate is expected to sign off on Murray’s plan late Friday or Saturday, depending on how long members choose to debate.

Murray said she and Ryan have exchanged phone calls since their breakfast meeting and have come to appreciate each other’s roles. “We have a relationship that understands that we both have an important job on our side to get budgets out – and all the challenges that come with that,” she said.

In a briefing with reporters earlier this month, Ryan said he thinks the two sides have an incentive to work together.

“We’re going to have to talk to each other to get an agreement about how to delay a debt crisis,” he said.

And at a Capitol Hill news conference last week, he added: “Look, I’m very pleased that Patty Murray is attempting to pass a budget because we haven’t seen that attempt in a long time. . . . If she can pass a budget, then we actually have a process out in the public for the nation to see that gets us going down the path of solving problems.”

Murray said her new assignment is a big switch from leading the Senate Veteran’s Affairs Committee, which she headed in 2011 and 2012. There, she said, Republicans and Democrats were eager to work together to advance the popular causes of veterans.