A majority of the Senate, including the Republican and Democratic leadership, urged President Reagan yesterday to make trade a high priority in his talks next month with Prime Minister Yasuhiro Nakasone.

The 53 senators, headed by Majority Leader Robert Dole (R-Kan.) and Minority Leader Robert Byrd (D-W. Va.), warned that attempts to paper over trade frictions with "stopgap solutions" would increase the chances of Congress passing protectionist legislation.

Congressional sources said the letter was sent to Reagan because of possible pressures within the administration to avoid embarrassing Nakasone with contentious issues during his visit here and before the seven-nation economic summit in Tokyo in June.

With a trade surplus of nearly $50 billion with the United States last year, Japan has become the focus of trade tensions in this country. While the Nakasone government won praise for its efforts since September to increase the value of the yen, its recent moves to bring its currency back down have raised concerns that "Japan is unwilling to assume the burdens and responsibilities incumbent upon the second-largest economy in the Free World," the letter said.

The senators expressed the hope that "ongoing trade discussions will not be subjected to 'political' solutions by those less concerned with results than with creating a congenial atmosphere for the meeting.

"Unless current negotiations can be resolved in a way that offers measurable improvements in Japan's trade performance, your administration's efforts over the past year run the risk of being dismissed as an exercise in politics."

The letter was sent to the president as U.S. and Japanese negotiators began talks here on one of the knottiest trade issues between the two countries -- charges by the Reagan administration and U.S. companies that Japan has used unfair tactics to block American semiconductor sales in its market and to gain a major share of sales here.

The Japanese are hoping for a solution before Nakasone comes here April 12 to meet with Reagan, and U.S. Trade Representative Clayton Yeutter has said he believes an agreement is possible as a result of this set of negotiations.

But Yeutter was reported to have told key lawmakers this week that the administration would not accept a solution that was not fair to the domestic high-technology industry, and said his office could release an investigation detailing Japan's unfair tactics in semiconductor trade before Nakasone arrives.

At the same time, the Cabinet-level Economic Policy Council headed by Treasury Secretary James A. Baker III decided earlier this week that Reagan should focus in his talks with Nakasone on basic issues involving the Japanese economy.