"That's going to be a hard conversation".
The top USA diplomat's visit to Ankara comes amid escalated tensions between the two North Atlantic Treaty Organisation allies over a series of disagreements, including human rights cases and the Syria crisis. In Amman, Tillerson is expected to sign a multiyear, multibillion dollar USA aid package with Jordan to shore up the relationship.
"Look, it's hard. The rhetoric is hot, the Turks are angry and this is a hard time to do business, but it's our belief that there are still some very fundamental underlying shared interests", the senior official said Friday.
"These are some of our closest partners but they are also partners with whom we are facing some of the toughest issues that we have to face in the region, whether its terrorism, whether it's the future of USA assistance, whether it's the final defeat of ISIS (the Islamic State militant group)".
Given the news, Tillerson has "not ruled out" adding a stop in Israel in order to help diffuse the situation, a State Department official said.
Tillerson finishes the trip in Ankara with surely tense talks with Turkish President Recep Tayyip Erdogan. Gulen has denied any role in the plot.
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The lack of trust between Washington and Ankara grew after Turkey started an air and ground offensive in Afrin, Syria against a Kurdish group known as the People's Protection Units, or YPG.
Tillerson is traveling through Jordan, Turkey, Lebanon, Egypt and Kuwait over the next five days, where he already planned to focus his discussions on the ongoing crisis in Syria. The U.S. officials said Tillerson will repeat warnings for Turkey to show restraint in military operations in Kurdish areas of Syria.
Turkey is one of his stops, where he will tackle bilateral ties and regional issues with senior Turkish officials.
The Secretary will lead the USA delegation to the Ministerial meeting of the Global Coalition to Defeat ISIS in Kuwait and participate in the Iraq Reconstruction Conference.
It will be the third Global Coalition meeting and the first since the Islamic State extremists were driven out of their erstwhile strongholds in Raqqa, Syria, and Mosul, Iraq, and mostly defeated on the battlefield throughout the two countries.
Tillerson will not be making any new USA assistance pledges at the Iraq Reconstruction Conference, the officials said.