The eyes of the world will be on President Reagan and Soviet leader Mikhail Gorbachev at the summit meeting in Moscow in late May, but their wives may steal some of the spotlight.

Nancy Reagan and Raisa Gorbachev have met on several social occasions during two previous summit meetings - Geneva in November 1985 and in Washington in December 1987. But their coolness to each other stands out in contrast to the growing warmth between Reagan and the Kremlin leader.During the four days the Reagans will spend in the Soviet Union, the two first ladies will be thrown together often, but this time Mrs. Gorbachev will have to play the gracious hostess.

Mrs. Reagan and Mrs. Gorbachev were reared in different worlds and have little in common. The Kremlin first lady is a university professor and a strong and vocal advocate of the Marxist-Leninist philosophy.

Mrs. Reagan, a former actress and Smith College graduate, has found herself on the defensive in their tea-time chats that escalated into some tense moments in their East-West dialogues.

In his new book "For the Rec-ord: From Wall Street to Washington," former White House chief of staff Don Regan recalled the rivalry between the two women at the Geneva summit meeting.

Describing Mrs. Gorbachev as a stylish, handsome, intense woman with an intelligent manner, Regan said the Soviet first lady dominated the dinner party given by the Reagans and kept the conversational ball rolling.

"Instead of cross-chat with Mrs. Reagan on palace housewifery," he wrote, "Mrs. Gorbachev is a highly educated woman . . . who did not hesitate to educate Reagan on Soviet policy."

"She was the mistress of her subject, an intellectual with a truly impressive grasp of Soviet policy ... entering into conversations down the table to express an opinon."

Regan wrote that as Mrs. Gorbachev continued her monologue, holding Reagan as a captive audience, Mrs. Reagan chafed. After the Gorbachevs left, Regan recalled that Mrs. Reagan said, "Who does that dame think she is."

They hardly hit it off when they first met in Geneva, and everything was downhill in Washington where Mrs. Gorbachev appeared to ignore Mrs. Reagan even when she visited the White House.