State Department sources said last evening that the United States is in the process of granting political asylum to Soviet conductor Maxim Shostakovich, the son of renowned composer Dmitri Shostakovick who died in 1975.

The 42-year-old conductor requested police protection for himself and his 19-year-old son in West Germany on Saturday, after completing a European Orchestra. The State Department sources said yesterday that the two "may well arrive in the next few days."

The asylum process began yesterday morning when National Symphony Orchestra music director Mstislav Rostropovich phoned Deputy Secretary of State William Clark to suggest that Shostakovich be granted asylum.

"I called my friend Mr. Clark not only because Shostakovich is a great artist," Rostropovich said yesterday, "but also because he is my friend. No, more like family. I have known him since he was 4 years old. He is the son of the greatest Russian composer, the greatest music teacher and my greatest friend. I felt I had to do this in memory of his father."

Last evening Clark confirmed his conversation with Rostropovich: "Slava called me," he said, "and said Shostakovich wanted asylum. We've been looking into it."

The State Department sources said that Shostakovich is expected to come to Washington. And, in the manner of surprises that Rostropovich has been known to unveil, this most recent Soviet defector may well appear on stage with the National Symphony.

Maxim Shostakovich has been one of the Soviet Union's most popular conductors, appearing often on television and in person. Two years ago he became embroiled in a controversy over a book purported to be the memoirs of his father. The book criticized the Soviet system, although in public, Shostakovich the composer was an outspoken defender of Marxism. Shostakovich the conductor said the book was not the work of his father, who died of a heart ailment in a Moscow hospital in 1975, seven years after illness forced him to resign as head of the Soviet Composer's Union.