Vice President Bush told Canadian leaders here today that President Reagan has "a driving motivated desire" to seek strategic arms reductions with the Soviet Union.

Bush, in a one-day visit to the Canadian capital, met with Prime Minister Pierre Elliott Trudeau and other members of his increasingly embattled government. The schedule included working talks through the day and ended with an official dinner for the vice president, his wife and the U.S. delegation.

Relations between the two countries have been strained by differences over economic, environmental, energy and now strategic weapons questions. Canadian anxiety over the Reagan administration's tough anti-Soviet rhetoric has coalesced since last fall. Local referenda in last fall's Canadian elections produced an overwhelming yes vote on the question of disarmament.

More recently, Canadian concerns on the issue have centered on U.S. hopes to test-fly cruise missiles over northern Canadian territory to check out their radar navigation systems. The unarmed missiles would overfly about 1,400 miles of Canadian soil.

Last month the two countries signed a framework agreement under which a separate pact on the cruise flights could be negotiated.

The Canadians, led by Trudeau and his deputy prime minister, Allan J. MacEachen, who is also foreign minister, pressed the Bush delegation for progress in Euromissile talks with the Soviets. MacEachen said in a press conference this afternoon, "We wanted to make clear our view that progress must be made if possible" in reducing strategic nuclear arms in Europe. "So far there has been no breakthrough."

The North Atlantic Treaty Organization agreed in December 1979 to deploy 108 Pershing II missiles and 464 ground-launched cruise missiles in response to a Soviet rocket buildup. Reagan has proposed the so-called zero option in which both sides would eliminate nuclear missiles from the European theater. Controversy over the NATO decision has grown as the time of deployment draws near. However, Bush said today that the recent West German elections in which Chancellor Helmut Kohl increased his strength indicates to Washington that NATO's two-track strategy of talking while preparing to add arms was winning renewed European support.

So far the Soviets have rebuffed the Reagan proposal. "There's been no breakthrough," MacEachen said. "It is clear that the zero option has not led to agreement. It hasn't succeeded, and we wanted to explore the necessity of offering alternatives while not abandoning the zero option."