Days after his arrest, the Istanbul police searched the home of a third employee, Mete Canturk, who has also worked for years at the Istanbul consulate. He was not at home at the time, and officers then visited his wife’s family home in northern Turkey and detained his wife and daughter, who were attending the funeral of his wife’s father.

The recent detentions prompted the United States visa suspension, an unexpected and drastic action that seems to have halted the arrest of a third employee. Since the suspension, the Turkish police have not pursued Mr. Canturk, who has continued to work every day at the consulate.

But his wife and daughter were detained for more than a week in Amasya. They were released Monday evening on the eve of the arrival of Mr. Cohen, the State Department official, but remain under judicial control and cannot leave the country.

The American action followed a similar turn in German-Turkish relations several months ago, when Germany reacted to the arrests of several Germans in Turkey by threatening economic sanctions and restricting support for Turkey’s application for membership in the European Union.

Turkish officials have indicated that they want the visa suspension to end.

“I don’t think that this is a problem that will drag on,” Mr. Erdogan said on Thursday. “This delegation is here to finish it. The talks are ongoing. My wish is for a result as soon as possible, leave the visa issue behind and that relations return to its normal situation.”

Yet, in other comments later the same day to a gathering of party supporters, Mr. Erdogan continued to rail against United States policy, in particular at perceived American reluctance to extradite Mr. Gulen.

“They think we are strategic partners and that we will obey everything,” he said of American officials. “Now they are asking me to give them some people. But first you should give me the one in the ranch,” a reference to Mr. Gulen and his Pennsylvania home.