Senate Democrats have floated the idea that the House-Senate Budget conference could reach an agreement if just one House Republican switched sides and signed off on a conference report, but don’t expect that to happen — the likeliest candidate, Oklahoma Republican Rep. Tom Cole, has no plans to do so.

Cole, the newly anointed chairman of the Legislative Branch Appropriations Subcommittee, would be the obvious choice as a member of the Appropriations Committee who wants to get to work on the nuts-and-bolts of a fiscal 2014 spending package.

He joined the other House cardinals in a Monday letter to the bipartisan leadership of the conference committee seeking two years of top-line spending levels to allow the appropriators and their staffs to get to work on a funding package for the rest of the fiscal year. The appropriators want their numbers by no later than the beginning of December.

The self-imposed “deadline” for the conference is not until Dec. 13. Because there are only seven House conferees, a single member could tip the scale in either direction.

Senate Democrats, led by Budget Chairwoman Patty Murray, D-Wash., insist that an agreement help pay for the offsetting of the sequester with revenue increases.

Cole had told Bloomberg TV that he personally could be open to tax code changes like closing the tax loophole regarding treatment of “carried interest,” as well as a repatriation holiday for bringing money back from overseas. Still, he threw cold water on the idea of breaking from his fellow House GOP conferees during a brief interview late last week.

“Look, I’m there to work with my colleagues and support my chairman, not to undercut them,” Cole said.

“I never rule any scenarios in or out. Look, we’re not gonna be at a position that the speaker’s not at,” Cole said. “He appointed all of us to this committee so, you know, I want to work with my colleagues and find a deal and we’re all good friends, we all know one another, we all serve on the Budget Committee together, so hopefully we can all find common ground and be in the same place.”

The meat ax cuts that go along with the sequester will hit Cole’s military-heavy Oklahoma district particularly hard if a deal isn’t reached.

When Senate Minority Leader Mitch McConnell visited House Republicans Tuesday morning, he had a clear message: Republicans should maintain their position in support of lower spending levels dictated by the Budget Control Act. That would mean a top-line level of $967 billion for fiscal 2014, perhaps unless there are other cuts on the mandatory side.

Cole noted the possibility (if not the likelihood) that leaders would get involved in a final agreement.

“Not everybody that matters in this agreement is in that room. So if there’s an agreement above our heads at the leadership level, I suspect that everybody in the room would reflect what leadership wants — well maybe not every single member, but predominantly would. If our leadership decided on something then I suspect we all would try to be supportive of that. On the other hand, if they were against it, it would be very unusual for someone to try to undercut their own leadership,” Cole said.