France's foreign minister said last week EU ministers would discuss Turkey on July 13 and said new sanctions on Ankara could be considered in addition to steps taken over Turkey's drilling in the Cyprus economic zone.
"If the EU takes additional decisions against Turkey, we will have to respond to this," Cavusoglu told a news conference with EU foreign policy chief Josep Borrell in Turkey's capital.
As relations deteriorated between the bloc and Ankara, the European Union imposed a travel ban and asset freezes on two people in February for their role in Turkey's drilling in Cyprus' maritime economic zone off the divided island.
Cavusoglu berated the EU for failing to fulfill promises and linking issues such as the Cyprus dispute and a 2016 migrant deal. He said Turkey will not allow itself to be held hostage by Greece and Cyprus and called on the EU to be an "honest broker".
Earlier this year, tens of thousands of migrants tried to cross into Greece via land and sea borders after Ankara said it would no longer stop them. The flow has slowed since then, but Cavusoglu said Turkey "will continue to implement its decision".
Cavusoglu also repeated a call for France to apologise after an incident between Turkish and French warships in the Mediterranean, which prompted Paris to request a NATO investigation.
Relations between the NATO members have soured over the Libya conflict, where Turkey supports the internationally recognised government and accuses Paris of backing the eastern-based forces of Khalifa Haftar who tried to capture Tripoli.
France denies backing Haftar's offensive on the capital, and accused Turkish warships of aggressive behaviour.