Soviet Foreign Minister Eduard A. Shevardnadze held nearly three hours of talks Friday at the State Department as U.S. officials anticipated a proposal that interim limits be placed on the most potent U.S. and Soviet long-range nuclear weapons.

The proposal by General Secretary Mikhail Gorbachev was designed to maintain momentum in arms control efforts as the Reagan administration nears the end of its term.It was sent to Washington before Shevardnadze held talks with Secretary of State George P. Shultz on a wide range of arms control measures, human rights and regional problems.

"We made some progress," Shevardnadze told reporters as he returned for a second day to the State Department.

He also said he had arranged for a visa for an Armenian emigre who wants to visit his ailing mother in Yerevan, the capital of Armenia. On Thursday, as Shevardnadze left the department, Ambarsum Khlgatian, 61, approached the foreign minister's motorcade to present a leaflet asking for assistance.

The protester was wrestled out of the way by security guards - until Shevardnadze surprised observers by getting out of the car to talk to the man.

Shevardnadze waved aside reporters' questions after Friday's meeting with Shultz. He planned to return later in the day after a meeting with President Reagan.

Meanwhile, Reagan told a political rally in Boca Raton, Fla., that the Soviets had negotiated with the United States because he took a tough stance toward Moscow.

Reagan sought to credit Vice President George Bush with contributing to a successful policy. "They knew we meant business," Reagan said.

The interim agreement proposed by Gorbachev would limit the number of warheads on all U.S. and Soviet long-range missiles. Shevardnadze did not present the idea formally during the first day of talks. But, he said, "we shall continue our discussions."

The U.S. side is willing to consider the proposal, but will emphasize that any limitations on nuclear weapons must be reinforced with strict verification measures to guard against violations, the officials, who spoke on condition of anonymity, said Thursday.

These include assurances there will be no interference with the monitoring of test flights of the strategic nuclear missiles that carry the warheads and have ranges of 3,500 miles and beyond.

After his meeting with Shevardnadze, Bush told reporters, "If my eyes look a little heavy, I got in at 2 o'clock this morning from Texas." He had attended a GOP fund-raiser in Houston Thursday night.

Shevardnadze said of Bush, "He has been involved in all high-level discussions here in Washington and our talks with the president."

The United States and the Soviet Union set goals for limiting warheads at the summit last December. Gorbachev wants to put those limits into effect on an interim basis, separate from other arms issues.

These would include a ceiling of 10 warheads on U.S. ground-launched MX missiles and eight on submarine-launched Tridents, while the Soviets would be held to 10 warheads on their SS-18 and SS-20 missiles, the official said.

Shevardnadze carried a range of other arms control proposals to Shultz along with a letter from Gorbachev to Reagan.

"The agenda is very broad, but we have little time," Shevardnadze told reporters.

He singled out two obstacles to a Strategic Arms Reduction Treaty (START) as included in the "full-scope negotiations" he was holding with Shultz. They involved restrictions to be imposed on the range of non-ballistic cruise missiles that can be launched by bombers and warships and ways of verifying that any agreed-upon cutbacks on their number are not exceeded.

Shultz met with Shevardnadze for 6 1/2 hours and afterward described their discussion of arms control issues and human rights as "very constructive and very detailed."

Negotiators in Geneva have been unable to resolve these problems, as well as related questions dealing with Reagan's Strategic Defense Initiative, the search for a space-based shield against Soviet ballistic missiles.