U.S. officials have proposed a mid-June summit between President Reagan and Mikhail Gorbachev in Washington and anticipate that the Soviet leader will spend about a week in the United States, including a possible visit to the president's home state of California.

"What Gorbachev actually visits here will reflect his own wishes, but the president would like him to see the diversity of America," a senior official said.

U.S. officials are working on the assumption that the second Reagan-Gorbachev summit would be held about June 15, but they do not yet have a firm commitment from the Soviets on an actual date, sources said.

Before Reagan and Gorbachev sit down again, two rounds of talks will be held by U.S.-Soviet arms-control negotiators in Geneva. Little progress is expected in the first round, beginning Jan. 16, because of Soviet preoccupation with the Soviet Communist Party congress in Moscow the following month.

But U.S. officials think that the second session could produce serious bargaining. U.S. officials do not at this time expect a new arms-control agreement at the Washington summit, but they think that the leaders of the superpowers may agree on some concrete steps that go beyond the general reaffirmations of arms control at last month's Geneva summit.

If this happens, the official said, it might be possible for the two leaders to sign an arms-control agreement at their third summit, at Moscow in 1987. The most tangible accomplishment of the Geneva meeting was the agreement of Reagan and Gorbachev to hold the two additional summits.

"The Geneva meeting was significant because the two leaders talked realistically at the first summit in six years," said a U.S. official. "Just repeating that process in Washington won't be seen by anyone as a success."

A U.S. official involved in the arms-control negotiations said that the Geneva summit may have had benefits that go beyond cosmetic easing of superpower tensions. The official said that Gorbachev learned that Reagan is serious about his missile defense plan, the Strategic Defense Initiative, and can't be pressured to abandon it.

At the same time, the official said, Reagan learned that the Soviets would not accept his idea of sharing missile-defense technology or agreeing to U.S. deployment of a missile-defense system.

The disagreement on missile defense remains a significant barrier to any arms-control agreement. Departing national security affairs adviser Robert C. McFarlane said Monday that he was optimistic about the prospect of an interim agreement limiting intermediate-range ballistic missiles in Europe, but the Soviets have insisted that the U.S. missile-defense project be limited before any agreement is reached.

It is also uncertain what impact the departure of McFarlane will have on U.S. arms-control efforts. During his two years as national security affairs adviser, McFarlane spearheaded efforts to achieve an administration compromise on arms control with only limited success. Some administration officials think that it will be much more difficult to achieve consensus with McFarlane gone.

Whatever happens on arms control, administration officials say that Reagan will try to give Gorbachev a different picture than he now has of the United States by taking him to farms, factories and residential areas to demonstrate the diversity of America and the productivity of its people. One official said the president would especially like Gorbachev to visit California but that this decision will be left to the Soviet leader.