Harvard is discussing with Soviet officials a possible stopover in Cambridge by Mikhail S. Gorbachev during his summit visit to the United States next month, University officials said today.

Although no formal invitation has been extended, the University made overtures to Soviet officials several weeks ago after Gorbachev announced plans to meet with President Reagan in Washington on December 7. News reports this week said that visit, which was originally to be limited to a brief two or three days in Washington, might be extended to as long as six days in order to permit the Soviet leader to tour other parts of the country.

If discussions with embassy officials are successful, this would mark the first ever visit by a Soviet General Secretary to any Western university, Harvard officials said.

But even if Gorbachev does decide to extend his visit, "chances are not even fifty-fifty" that Harvard would be granted a spot on the leader's cramped itinerary, said University Marshall Richard M. Hunt, who is handling the arrangements. "All the other institutions are providing him with opportunities to visit."

He said that Soviet officials have adopted a "don't call us, we'll call you" stance on a Harvard visit. Embassy officials could not be reached for comment yesterday.

A Gorbachev visit would be similar to that by Indian Prime Minister Rajiv Gandhi 3 weeks ago, Hunt said. He said that the Soviet leader would likely come for an afternoon and evening to give a speech and tour the University. The schedule would also include a closed-door meeting with President Bok.

Gorbachev may meet with faculties from the Slavic and Government departments and the Russian Research Center, Hunt said. He said the University would also attempt to arrange a discussion with students, but said that no details have been worked out.

University officials said that in addition to Harvard's stature, the large number of high-tech firms in the area might induce the technology-minded Gorbachev to visit Harvard.

"For Gorbachev it would make sense to come here. It would make sense to visit the Route 128 corridor" where much of the high-tech industry has gravitated in recent years, said Marshall I. Goldman '56, director of Harvard's Russian Research Center.

A Boston stop in the Route 128 area would take the place of a visit to California's Silicon Valley, which Reagan had reportedly urged. Goldman said however that a California trip is less likely because of the travel time involved.

The Soviet leader's trip was originally scaled back because of concerns that a prolonged visit would be a political asset to President Reagan, according to published reports. Gorbachev's visit was also limited because Soviet officials felt that he could not be adequately protected in a multi-city tour.

Security Worries

The same security worries would likely constrain Gorbachev's appearances here. "It would obviously be extremely difficult, if not a nightmare, if we would have to provide for the leader of a superpower," Hunt said.

Goldman said he thought Gorbachev, who is scheduled to sign an arms control agreement with President Reagan, would be politically wise to come to Harvard because of favorable publicity in the Soviet Union.

"It would play very nicely at home on Soviet TV," Goldman said.