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Garrett and Campbell both said they have been giving thought to what the position would entail.
"You're going to have a President Romney coming in, under that scenario, who has said he wants to balance the budget within a 10-year window. ... I'm all about that," Garrett said. "Whoever takes that position will be in a good spot to work with Romney and Ryan to achieve the Romney objective of balancing it far sooner than anything we've wanted to pass before."
Campbell said the role, in his eyes, would be picking up where Ryan leaves off.
"Paul Ryan has started a bunch of initiatives in the Budget Committee which have not been finished. The job, to me, would then be to finish what Paul Ryan started," he said.
One thing is clear, though. Whoever takes the committee helm will have big shoes to fill.
"It'd be a tough shadow to live in. Whoever gets it is going to have to follow Paul Ryan. Good luck," said Rep. James Lankford (R-Okla.), a committee member.
Indeed, staffers say none of the contenders measure up to the charismatic conservative celebrity that Ryan has become.
Though Garrett has the know-how to herald a budget and seems to be the favorite, staffers said the soft-spoken and unassuming Member would hardly be the front-and-center personality that Ryan is.
But one leadership aide noted that having a prominent committee chairman would be less necessary if Ryan is leading the charge from the White House.
Still, Garrett has occasionally been a thorn in GOP leaders' side, pushing for a budget lower than $1 trillion when Speaker John Boehner (Ohio) preferred a higher number.
"You've got to be somebody who is willing to pass serious budgets that work. The things they were asking for weren't viable," a GOP aide said.