The new Soviet leader, Yuri Andropov, met privately today with Vice President Bush and Secretary of State George P. Shultz in what appeared to be a significant attempt to halt two years of steady deterioration in Soviet-American relations.

Before leaving Moscow tonight, Bush said that the 30-minute meeting was "frank, cordial and substantive" and that it gave "both sides the opportunity to exchange views on the state of their relations."

While not characterizing the talks, the official Soviet news agency Tass said the meeting involved "a short exchange of opinions on the principal questions in Soviet-American relations."

The Soviet account of the meeting quoted Andropov as telling the Americans that his country "is ready to build relations with the United States on the basis of full equality, noninterference, mutual respect in the interests of the peoples of the two countries and normalization of the international situation."

Tass quoted Andropov as expressing "gratitude" to the Reagan administration "for the respect shown for the memory of Leonid Brezhnev." The Soviet leader, who died last Wednesday at age 75, was succeeded by Andropov on Friday.

Senior U.S. officials traveling with Shultz would not discuss the talks.

Andropov was accompanied by Foreign Minister Andrei Gromyko and Andrei Alexandrov, national security assistant to Brezhnev. Shultz and Bush were accompanied by U.S. Ambassador Arthur A. Hartman.

With U.S.-Soviet relations at one of the lowest points in years, today's meeting appeared to be a step in the direction of arresting their further decline and preventing them from deteriorating to the point that they would not be receptive to basic changes.

In the last weeks of Brezhnev's life, the Soviet Union forecast a counteroffensive in the fields of propaganda and weapons development that appeared to be pushing Soviet-American relations to just that point.

In his statement, Bush quoted President Reagan as saying that the United States has "strong desire" to work "toward an improved relationship with the Soviet Union."

"This is our purpose and our policy," Bush said. The president is ready, he added, "to conduct relations with the aim of expanding the areas where our two nations can cooperate to mutual advantage.

"Human rights, arms reductions, peaceful solution of regional problems, peace and freedom for all nations are the goals we seek. The achievement of these aims is the challenge our two great nations face," Bush said.

While it appears that no breakthrough is to be expected soon, the Soviets clearly have welcomed Reagan's recent conciliatory gestures but continue to be wary of American intentions.

In Washington, White House Deputy Press Secretary Larry Speakes said there was no indication that a summit between Reagan and Andropov would be held anytime soon, The Associated Press reported. Speakes said that the criteria for a summit would be whether it would achieve something and that the White House has "not seen anything to indicate that we are closer to a summit."

Diplomats here said Andropov's meetings with the Americans and the leaders of West Germany, India, Afghanistan and Pakistan all in one afternoon were designed to demonstrate that he is not a temporary figure and that Moscow has a new and robust leadership.

Such a schedule after the long funeral ceremonies and a speech stand in sharp contrast to Brezhnev's work schedule. The late Soviet leader, particularly after his illness last March, physically was unable to do more than a couple of hours work each day.

Immediately after the funeral, Andropov made a gesture obvious to all in the St. George's Hall in the Kremlin by singling out Chinese Foreign Minister Huang Hua for special attention. He chatted amiably with Huang and shook hands several times to delay the line by four minutes.

Huang has delayed his departure for Peking and is scheduled to meet Gromyko Tuesday. Chinese sources said it was likely that Huang also would meet with Andropov.

The new Soviet leader made another unusual decision today by foregoing an expected meeting with the leaders of Warsaw Pact countries. With the exception of Hungary's Janos Kadar and Bulgaria's Todor Zhivkov, all of them departed for home today.

The only Communist leader to gain a private, 30-minute meeting with Andropov was Babrak Karmal of Afghanistan, where some 100,000 Soviet troops are underpinning his government.

Some diplomats suggested that this was a signal that Andropov would like to deal with the Afghanistan problem. The Soviet invasion of Afghanistan in December 1979 was the starting point for a rapid deterioration in U.S.-Soviet relations.

Andropov's discussions with President Mohammed Zia ul-Haq of Pakistan and Indian Prime Minister Indira Gandhi also suggested that the Afghan problem is a high priority on the new leadership's agenda.

Coinciding with the Soviet-American meeting today was the arrival here of a huge delegation of U.S. corporate executives and the scheduled arrival of a congressional delegation led by Sen. Robert J. Dole (R-Kan.).