Secretary of State John Kerry said Sunday he told Turkish Foreign Minister Mevlüt Çavuşoğlu that it was irresponsible for his country to accuse the U.S. of involvement in Friday’s coup attempt.

In three phone conversations with Mr. Çavuşoğlu over the weekend, Mr. Kerry said he urged Turkey to play by the rules and follow international legal standards as it pursues its investigation. The U.S. is waiting for Turkey to present a formal extradition request for a Fethullah Gülen, an exiled cleric living in Pennsylvania who Turkey...