Turkish officials raised concern about a host of issues surrounding the coalition, including the safety of 49 Turkish diplomats who have been taken hostage by ISIS, and whether the growing international effort to arm Kurdish fighters in Iraq against ISIS could embolden Kurdish militants in Turkey who have been seeking autonomy for the country’s largely Kurdish southeast. Turkish Kurds with the P.K.K. have fought with Kurdish pesh merga fighters in northern Iraq against ISIS. Turkey is also grappling with an influx of more than 800,000 Syrian refugees — the largest Syrian refugee population after Lebanon’s.

Speaking to reporters after meeting with Mr. Erdogan on Monday, Mr. Hagel said that Turkish officials had expressed to him their concern about the P.K.K. But, he added, “They didn’t indicate to me in any way that they saw the P.K.K. as a more significant threat than ISIL,” using an alternative acronym for ISIS.

The Obama administration wants Turkey to crack down on the flow of foreign fighters who have used the country as a transit point to Syria to join militant groups fighting there. The United States also wants to be able to use Turkish military bases to begin operations, including airstrikes, on ISIS targets in Iraq and Syria if President Obama decides to attack inside Syria.

“By geography, Turkey is going to be absolutely indispensable to the ongoing fight against ISIS,” a senior defense official said, speaking on condition of anonymity. “Because of just where they sit, the access we currently already have militarily and the cooperation we have militarily.”

Mr. Erdogan, at the NATO summit meeting in Wales last week, met with Mr. Obama and said Turkey would become the only majority Muslim country so far to join what Mr. Hagel termed a “core coalition” of 10 countries fighting ISIS. Mr. Obama said after the meeting that he would welcome Turkey’s participation in his coalition.