Walsh is the Democratic frontrunner in the race to replace Baucus, and his appointment will give him the advantage of incumbency in name recognition and fundraising, the Washington Post said.
Baucus was confirmed as the next ambassador to China Thursday. President Obama nominated him to the diplomatic post late last year several months after he announced he would retire at the end of his current term.
Former Lt. Gov. John Bohlinger already announced he would seek the Democratic nomination.
Rep. Steve Daines is the likely GOP nominee, the Post said.