Minayev made his debut for USSR on 20 March 1976 in a friendly match against Argentina.

He played in the quarterfinals of the UEFA Euro 1976 and participated in the qualifiers for the 1978 FIFA World Cup; however, the USSR did not qualify for the final tournament in either competition.

In the first half of the 1980s, he was in a long-term relationship with actress Natalya Gundareva.

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