His statement comes ahead of his trip to the US, and he said he would discuss the relations with US officials during the visit.
Washington has repeatedly said that Turkey must give clear evidence of Gulen’s link with the attempted coup before discussing any possible extradition. Lawyers have said the process, if launched, may take years.
Cavusoglu also said Turkey may remove some ambassadors linked to the coup attempt.
Cavusoglu warned the EU that it “can’t threaten Turkey.”
“We’re protecting European values. EU officials often say what they think, but when it turns into a threat, it becomes counter-productive.”
Earlier on Monday, EU Commission President Jean-Claude Juncker told broadcaster France 2 that he believes that“Turkey, in its current state, is not in a position to become a member any time soon and not even over a longer period."