Pompeo told reporters on the sidelines of an Asian security forum in Singapore that the two countries would continue to work with each other in the framework of the alliance and on other matters.
Cavusoglu, however, said both the countries have agreed to work together.
Secretary of State Mike Pompeo pressed Turkey on Friday for the release of detained American pastor Andrew Brunson, warning that the issue was "very serious".
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 USA pastor, which has already resulted in Washington slapping sanctions on Ankara.
The sell-off, which also hammered local stocks and Turkey's debt risk profile, reflected deepening investor concern over tensions with the United States, a North Atlantic Treaty Organisation ally and major trading partner.
Cavusoglu, in comments to Turkish broadcasters, said he and Pompeo had also discussed potential joint steps the two countries could take regarding Syria's Idlib and Manbij.
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Also caught in the spat is a plan for the United States military to deliver new F-35 fighter jets to Turkey.
"Since the beginning we said that a solution can not be reached by using threatening language and sanctions", the minister added.
NASA scientist Serkan Golge, a U.S.
Brunson, who led a Protestant church in the Aegean city of Izmir, denies the charges. "President Trump has made it abundantly clear that the United States expects Turkey to release him immediately".
The US Treasury implemented the sanctions under the 2016 Global Magnitsky Act, named after Russian lawyer Sergei Magnitsky, who died in a Moscow jail, and which allows the US to sanction foreign officials implicated in rights abuses.
Gulen, who has lived in self-imposed exile in the United States since 1999, has denied any role in the 2016 coup attempt.