The lawyer has taken part in an array of high-profile cases, including representing the family of murdered journalist Anna Politikovskaya, a fierce Kremlin critic, and the families of sailors who died in the Kursk nuclear submarine disaster in 2000.
By seeking asylum, Kuznetsov is "striving to secure himself from lasting and steadfast criminal prosecution," his lawyer Robert Zinoviev told The Associated Press.
Kuznetsov is one of many Russian lawyers to face official pressure after or while working on sensitive high-profile cases or on behalf of clients whom the government perceives as its opponents.
Kuznetsov had appealed to the court to rule on whether the Federal Security Service, the successor of the KGB, violated the rights of his client, a former member of the upper house of parliament, by tapping his phone conversations without court authorization.
His client, Levon Chakhmakhchyan, was arrested in February on suspicion of accepting a $300,000 bribe. Security service agents claim to have caught him in a sting operation.