Cavusgolu and Pompeo met on the sidelines of a meeting of regional ministers in Singapore.
The Turkish foreign ministry called on the United States to reverse its "wrong decision" on Thursday after Washington announced sanctions on Abdulhamit Gul, the minister of justice, and Suleyman Soylu, the minister of interior.
On Friday, the two leaders agreed to work together to resolve tensions between the two countries.
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That's according to three White House officials Tuesday, who were not authorized to speak publicly about private conversations. The president later mentioned his top aide's service on Twitter with a photo of the two men smiling and standing side by side.
The Turkish lira on Wednesday slumped to record lows of 5.0 against the dollar as the United States hit Turkey's justice and interior ministers with sanctions over the case of an American pastor on trial for terror-related charges.
Top diplomats from Turkey and the United States agreed on Friday to work towards a solution to the diplomatic spat that has erupted over the arrest of a U.S. pastor, which has already resulted in Washington slapping sanctions on Ankara.
Foreign Minister Mevlut Cavusoglu, who is set to meet US Secretary of State Mike Pompeo in the next few days, also warned that the move "will not go without response".
"Brunson needs to come home, as do all the Americans being held by the Turkish government", Pompeo told reporters before the meeting.
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That effort included sifting through data from her cellphone, social media accounts and a Fitbit , a physical activity tracker. Jack has said that he believes Tibbetts would have had both her fitness tracker and cellphone when she went out for her jog.
Speaking to Turkish journalists after the meeting, Cavusoglu also described the discussion with Pompeo as "constructive", but added: "Nothing can be achieved through threatening language and sanctions".
Turkey has said the sanctions were unacceptable. They have been at odds over numerous matters, including military activity in northern Syria and Turkey's plans to purchase an advanced air defense system from Russian Federation.
The Foreign Ministry strongly protested against the U.S. Treasury Department's decision, calling on the U.S. administration to reverse its "wrong decision". He was arrested in December 2016 following a failed coup on charges of "committing crimes on behalf of terror groups without being a member" and espionage.
Turkish President Recep Tayyip Erdogan has rejected the USA demands, saying his government won't back down and is willing to "go its own way" if the US acted.
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It added that the US' "aggressive attitude serves no goal", and vowed that it would respond in equal measure. Ankara blames Gulen for the coup attempt, while the cleric denies involvement.