The bill then would go to the White House, which has expressed modest opposition, leading its proponents to hope that President Reagan would sign it. A White House spokesman declined to state the President's intentions today.

Asked if he thought President Reagan would sign the measure, Senator Matsunaga said he was optimistic after talking with the White House chief of staff, Howard Baker. Some Conservative Opposition

Those voting in favor included senators of all sectors of the two parties. The opponents were largely conservative Republicans, though there were several exceptions, such as Senators Alan Simpson of Wyoming, Ted Stevens of Alaska and Orrin G. Hatch of Utah. The senators from New York, New Jersey and Connecticut all voted in favor.

Senator Matsunaga, a Democrat who was wounded twice after participating in the Anzio landing in Italy in World War II, said during the debate, ''A stigma has haunted Japanese-Americans for the past 45 years.''

After the attack on Pearl Harbor, Mr. Matsunaga was detained with other Japanese-Americans at Camp McCoy, Wis., although he was a second lieutenant in the Army and a company commander. He and other internees petitioned President Roosevelt to give them a chance to serve. They were allowed to form the 100th Infantry Battalion, which became one of the most highly decorated in American military history.

Mr. Matsunaga's Democratic colleague from Hawaii, Daniel K. Inouye, served in the 442d Infantry Regimental Combat Team, a group of Japanese-American enlisted men who served under white officers. He lost his right arm fighting in Italy and was awarded a Bronze Star and two Purple Hearts. Sorrow Across the Years

Tears rose in Mr. Matsunaga's eyes as he recalled the internment camp incident. As Mr. Matsunaga faltered, Senator Stevens came to his rescue while Senator Matsunaga regained his composure and sipped a glass of water.