Speaker John Boehner is officially neutral but privately supporting McMorris Rodgers. Paul Ryan, the returning chairman of the Budget Committee whose profile rose enormously during his vice presidential run, is asking colleagues to back Price.
McMorris Rodgers is the highest ranked woman in the House Republican conference, and a loyal leadership ally who has earned their trust as the conference's vice-chair. A surrogate for Mitt Romney's presidential run, she isn't known as a hard-line ideologue but has cultivated a conservative voting record in recent years.
Price, meanwhile, is the favorite of the GOP's conservative wing. A former chairman of the deeply conservative Republican Study Committee, he has been backed by outgoing conference chairman Rep. Jeb Hensarling (TX), who is expected to chair the Banking Committee next year.
Those who support McMorris Rodgers say she's in good shape to win.
"I think the Congresswoman is in a very strong position looking to tomorrow and has the support of everyone on the leadership table, except Price and Hensarling," a senior Republican aide told TPM, adding that she's better positioned to advance the GOP's message among women in the wake of the party's defeat last week.
Will it be a close race?
"Not really," the aide said. "I think the vote counts that we have show that she'll win pretty handily."
On Tuesday morning, Ryan sent out a "dear colleague" letter building support for Price: "I am asking you to join me in supporting Congressman Tom Price to be our next Chairman of the House Republican Conference," he wrote, calling Price "uniquely qualified for the job" and noting that he was "instrumental in drafting our House Republican budget."
"No one will work harder than Congressman Price at building a strong and compelling communications and policy strategy that reflects the will of our membership and the needs of the nation," Ryan wrote.
The GOP aide disputed a key part of the case for Price -- that he's more conservative. "That dog's not hunting," the staffer said. "If you actually look at the voting records of Price and McMorris Rodgers there's not much difference there."
Ellen Carmichael, a spokeswoman for Price, declined to discuss the internal jousting.
"As Chairman Price has always said, he believes this decision should be made through discussion and consultation with his colleagues, not through the press," she told TPM. "Despite there being interest to conduct this campaign in the media, Chairman Price, out of respect for his fellow Members, is focusing his efforts instead on one-on-one conversations with them and developing a real plan of positive, principled solutions for our Conference."
Leadership fights are notoriously difficult to handicap -- but the issue will be settled shortly after the GOP conference meets at 2 p.m. Wednesday.