Writing about a recent interview that Cavusoglu gave to German newspaper Frankfurter Allgemeine Zeitung, the Austrian foreign minister tweeted, "I do not accept Cavusoglu's ultimatum and turn it back. The EU cannot proceed with dependencies and must protect its borders."
In the interview, Cavusoglu had said, "We [Turkey] will have to take a step back from the readmission deal and March 18 agreement if visa-free travel is not granted to Turkish citizens."
He added that Turkey is expecting an exact date from the EU for visa-free Schengen zone travel and the EU should not see this as a threat.
Replying to Kurtz's tweet, Cavusoglu later said, "Das ist nicht [This is not] ultimatum Sebastian, it is my opinion. Freedom of expression, my friend."
The Schengen area is a 26-nation region in Europe where passports are not needed.
The readmission agreement between Turkey and the EU was signed in 2013 and went into effect in 2014.
The deal enabled EU to return illegal migrants who traveled through Turkey back to Turkey in order to be processed before they are sent back to their country of origin. Citizens of Iraq and Syria, which make up the majority of illegal migrants, are not part of the deal.
The March 18 deal aims to discourage irregular migration through the Aegean Sea by taking stricter measures against human traffickers and improving conditions of nearly 3 million Syrian refugees in Turkey.
Also under the deal, EU leaders agreed to cut visa requirements for Turkish citizens and accelerate Turkey's EU membership bid.