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.
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.
On Wednesday, Washington moved to block the assets of Turkish Justice Minister Abdulhamit Gul and Interior Minister Suleyman Soylu in the United States and prohibit American citizens from "engaging in transactions with them".
"I can say that it was a very constructive meeting", stated Cavusoglu.
Also caught in the spat is a plan for the United States military to deliver new F-35 fighter jets to Turkey.
The United States and Turkey have agreed to work closely to resolve issues between them, Turkish Foreign Minister Mevlut Cavusoglu said on Friday, following what he said was a "constructive" meeting with Secretary of State Mike Pompeo. But he said threats would not work.
On the sanctions, he said: "I hope they'll see this for what it is, a demonstration that we're very serious".
Mr. Brunson had denied all charges and his case has become a cause celebre for USA evangelicals and the Trump administration.
Pastor Brunson was arrested in Izmir, Turkey in October 2016, and was accused without any evidence of helping terrorist groups and of engaging in espionage.
Before his move home last week, Brunson, who led a Protestant church in the Aegean city of Izmir, had spent almost two years in jail.
But Brunson's case has resonated with U.S. President Donald Trump and more particularly with Vice President Mike Pence.
The former prime minister's position echoes that of the Turkish government.
The Erdogan government has vowed to retaliate against the USA sanctions, announced Wednesday, but has not revealed what they will be. Ankara blames Gulen for the coup attempt, while the cleric denies involvement.