U.S. officials don’t expect to extradite an imam Turkey blames for masterminding a failed coup because they aren’t convinced by the evidence Ankara has presented so far and are troubled by threatening public statements from Turkish officials, according to people familiar with the discussions.

U.S. and Turkish officials have privately discussed scenarios under which Fethullah Gulen might be extradited, but American authorities have yet to be persuaded there is a valid case for extradition, these people said. Mr. Gulen, who...