Nancy Reagan has invited Raisa Gorbachev to a tea and private tour of the White House in response to a "special request" from the wife of Soviet leader Mikhail Gorbachev to see the home of America's presidents.

A spokeswoman for Mrs. Reagan said yesterday that the White House learned of Mrs. Gorbachev's interest from U.S. Ambassador to Moscow Jack Matlock. On Tuesday, Mrs. Reagan invited Mrs. Gorbachev to join her the afternoon of Dec. 9, the second day of the third Reagan-Gorbachev summit meeting, and expressed the hope they would be able to spend "some private time together."

Elaine Crispen, the first lady's press secretary, said Mrs. Reagan wants to show Mrs. Gorbachev "the various priceless paintings and objects we have here."

Mrs. Gorbachev already has seen one of those paintings, Whistler's "Nocturne," a gift from the late W. Averell Harriman, who was U.S. ambassador to the Soviet Union. Normally, it hangs in the Green Room at the White House. Currently part of a Smithsonian-sponsored traveling exhibition in the Soviet Union, the painting drew a gasp of pleasure from Mrs. Gorbachev when she visited the show this week in Moscow, noting that "there is a sense of music and of dreams about it. Dreams, fog, night. It's unusual."

Crispen said Mrs. Reagan also wants to show Mrs. Gorbachev the White House reception rooms used on state occasions, which the public sees on daily tours of the mansion. But the first lady also plans to show Mrs. Gorbachev the family quarters on the upper floors, which the public never sees.

Crispen said the Gorbachevs' schedule for their Dec. 7-Dec. 10 visit here is still being prepared, so no definite time has been set for the tea. "We'll have to be a little flexible," Crispen said.

The get-together will be the first by Mrs. Reagan and Mrs. Gorbachev since the 1985 Geneva Summit, where they also met for tea. Mrs. Gorbachev caught the White House by surprise last year when she turned up, along with her husband, at the Reykjavik superpower meeting. Mrs. Reagan, unable to change a prior commitment at the last minute, stayed home. She later told reporters it had been the Americans' understanding from the beginning that she and Mrs. Gorbachev would not make the trip.

The text of Mrs. Reagan's letter: "Dear Mrs. Gorbachev,

"My husband and I look forward to welcoming you and the general secretary to Washington in December.

"Although current plans indicate that we will participate jointly in various official functions over the course of the visit, I am hopeful we can schedule some private time together as well.

"Because of your interest in art and historic restoration and your special request to tour the White House, I would like to extend my personal invitation to have you join me for a private tour and tea on Wednesday afternoon, the ninth of December.

"Knowing of the limited duration of your stay and the hectic schedule you will be keeping during your visit to Washington, I sincerely hope that we will be able to spend this private time together.

"Sincerely,

"Nancy Reagan."