The Washington rumor mill has begun to grind out names of possible choices for the new president's secretary of Interior.

The new Interior secretary will be the Bush Cabinet member with the most influence over the Mountain West, responsible for public lands, national parks and monuments, and reclamation projects like the Central Utah Project.So far, two Utahns, Richard Richards and L. Ralph Mecham, have been among names that have circulated as possible replacements for President Reagan's secretary, Donald Hodel.

Richards, chairman of the Republican National Committee in Reagan's first term and now a Washington lawyer, would like that job or an ambassadorship, rumors say, despite protestations he has made to the contrary.

Mecham, once administrative assistant to former Sen. Wallace F. Bennett, is director of the administrative office of the federal courts. He served as federal co-chairman of the Four Corners Regional Commission, then as a vice president of Arco Corp. He is well-regarded in Washington, but his stint with Arco, a mining company, taints him with some environmental groups.

The Sierra Club and other environmentalists would like Republican environmentalist Nat Reed, whose name floated here earlier this week, named to the Interior post. Reed's appointment to the department most involved with the West, however, would be considered a betrayal by Western Republicans, and by mid-week his name was being bandied about for the Energy portfolio instead.

The next most popular choice of the Sierra Club would be retiring Republican Sen. Dan Evans of Washington state. But Sen. James McClure, R-Idaho, a Bush confidant, has told Bush that Evans is not acceptable to the West, either.

Also being considered is New Mexico's Republican Gov. Garrey E. Carruthers. Carruthers would be more than acceptable to Western Republicans like Sen. Jake Garn, R-Utah, but his term in Sante Fe still has two years to run.

Carruthers met Tuesday with Bush transition officials, heightening the rumor that he is under consideration. He was to have been at a nuclear waste meeting in Salt Lake City but canceled that session to talk to the Bush aides.

Former Utah Gov. Scott Matheson, though a Democrat, would be acceptable to most Western conservatives, and Bush has dropped hints he might put at least a token Democrat in the Cabinet. But Matheson's refusal of a job as czar of nuclear waste disposal under Reagan in order to keep his options open in a potential Dukakis administration may have scotched Matheson's chances.