Several people are eyeing the chairmanship of the Federal Communications Commission, according to sources at the agency that regulates broadcast and phone companies, despite the fact that Chairman Mark S. Fowler's term will not expire until 1986.

Two of the five commissioners who preside at the agency -- Dennis R. Patrick and Mimi Weyforth Dawson -- are rumored to be contenders for the coveted chairmanship that will ultimately be decided by President Reagan.

Commission insiders say Fowler, previously rumored to be considering a return to private law practice, is now open to reappointment. "He has announced to several people he is toying with the idea," said one source close to Fowler.

"I have a three-word comment: Premature, I hope," said Fowler.

But the name of David J. Markey, assistant secretary of Commerce for telecommunications and information policy at the National Telecommunications Information Administration arm of the Commerce Department, has passed the lips of more than one inside source as a candidate for the job.

"David Markey has been campaigning very hard for it," said one FCC source.

"I'm flattered for people to even mention it," said Markey, who said he was not a candidate for the chairmanship. "I hope Mark Fowler stays beyond his term; he has done a good job. When he leaves either Mimi Dawson or Dennis Patrick would do a good job," he said.

That Patrick and Dawson both wouldn't mind being the Republican chairman of the agency has been plain to agency insiders for quite some time. "They are both talented and ambitious people and would both like to be chairman," said one source. "They joke; they'll say, 'If I were chairman I would do this differently.'"

Both commissioners try hard to outdo the other at regular commission meetings. "When one has to make a point, the other tries to identify themselves with a given position and outdo the other," said another source. "It's the Annie-get-your-gun thing."

"I think it's fair to say Mimi has been campaigning for chairman for quite a few years before Dennis came along -- until Dennis was named and reconfirmed she was the only other available Republican," the source said. Both commissioners are described as ambitious and hard-working, but Patrick is picking up on Dawson's technique of accessibility to the many groups that maintain contact with the commission, the source said.

"Dennis is making an effort to do something Mimi has traditionally done . . . . He is more accessible to everybody."

As far as being FCC chairman, "Mimi and I kid about it sometimes," said Patrick. " . . . It will be the president's choice and it's too early to start to speculate so I am not going to comment," he said.

Dawson said she had not addressed the question. "We have a chairman here who is doing a very good job," she said. "There isn't a battle and I enjoy my job and any decision is going to be made by the president."

If political contacts matter, both Republican commissioners have a leg up, say insiders. Dawson is a veteran Capitol Hill staffer who spent five years working for Sen. Bob Packwood (R -- Ore.) and coordinated activities of the National Republican Senatorial Conference. Patrick, a lawyer, confirmed recently for another term, was associate director of presidential personnel at the White House from 1981 to 1983. Patrick also clerked for former interior secretary and longtime Reagan troubleshooter William P. Clark when Clark was a California Supreme Court justice.

"Mimi's strength is her managerial prowess," said another source. " . . . and she has an excellent relationship and sensitivity to the Hill." On the other hand, "Patrick takes the time to do the analysis," the source said. "His White House connections and the fact that "Clark was his 'godfather' . . . can't hurt," the source said.

But, it could turn out the appointment goes to some third party as yet unmentioned. "Where is the political payoff for the administration in appointing someone to a plum position like that if you've already, as the president, appointed these people as commissioners?" the source said, "How do you benefit by appointing a Mimi Dawson or Dennis Patrick and gain political capital?