President Reagan offered to travel to Moscow for a second summit meeting with Mikhail Gorbachev if the Soviet leader would come to Washington first, a White House official said yesterday.

The official made the disclosure in illustrating Reagan's willingness to go the extra mile to arrange a U.S.-Soviet summit, which the two nations formally announced yesterday for Nov. 19-20 in Geneva.

Secretary of State George P. Shultz, in a news conference following the announcement in Washington and Moscow, said Reagan and Gorbachev have agreed in correspondence that "they would like to see a more constructive relationship emerge from the meeting." Shultz predicted that "an extensive preparatory effort" between now and November would improve the chances that "we won't have a situation where two people just get together and say hello."

Shultz said he expects to meet the new Soviet foreign minister, Eduard Shevardnadze, in Helsinki at the end of this month and at the United Nations in September.

Saying that the administration has "no illusions" about the differences in policy and political systems between the two nuclear superpowers, Shultz added that "it's important for us to redouble our joint search for ways to reduce tensions, lower the dangers of confrontation and conflict." He said Reagan hopes for "progress in that direction" as a result of the meeting. Reagan seemed to shy away from a summit meeting in his first term, when Leonid Brezhnev, Yuri Andropov and Konstantin Chernenko held the top Soviet post in rapid succession. But the president took the initiative in March of this year by sending Vice President Bush to Moscow for Chernenko's funeral with a letter inviting his successor, Gorbachev, to a summit.

Reagan's proposal was that Gorbachev come to Washington, since the last summit meeting on the soil of either country was the 1974 meeting in Vladivostok of Brezhnev and President Gerald R. Ford.

Gorbachev within a few weeks responded positively to the idea of the meeting, but left the time and place to be arranged through diplomatic channels.

Throughout most of the spring, the U.S. belief was that Gorbachev would come to the United Nations this fall to commemorate the world body's 40th anniversary, which would have made a meeting with Reagan in New York or Washington possible. In May, however, the Soviets passed word that Gorbachev was "too busy" for such a trip and, according to U.S. sources, suggested informally that Reagan come to Moscow.

It was evidently at this point that Reagan offered to travel to Moscow, possibly next year, if Gorbachev came here first.

Shultz's meeting with then-Foreign Minister Andrei Gromyko in Vienna May 14 reportedly moved the summit discussion forward, though there was no agreement then on a time and place. A State Department official said this week that a summit meeting was "pretty well agreed" on by the time the two ministers left Vienna.

Once it became clear that neither Washington nor Moscow was acceptable as a meeting place at this time, the discussion began about a neutral site. U.S. sources said the Soviets suggested Helsinki or Vienna, while the United States preferred Geneva.

Several considerations reportedly went into the choice. Helsinki is close to the Soviet border and would raise echoes of the 1975 Helsinki accords, which Washington has accused Moscow of ignoring. Vienna was more acceptable to Washington, but was the site of Brezhnev's meeting with President Jimmy Carter for the signing of the SALT II treaty, which Reagan opposed.

Geneva was the U.S. choice for symbolic and logistical reasons. Extensive offices and secure communications are in place there because of the arms talks and other frequent international discussions.

The United States suggested Geneva and the November dates a few days ago. The Soviets intimated earlier that November might be acceptable to Gorbachev. Soviet Ambassador Anatoliy Dobrynin delivered Moscow's positive reply to Shultz on Monday.

At the same time, the Soviet ambassador in Paris reportedly passed word of Gorbachev's willingness to make an official visit to France Oct. 2-5. This first trip by Gorbachev to a western country since becoming general secretary of the Soviet Communist Party was announced yesterday in Moscow and Paris.