Speaking after Eid prayers in Istanbul, President Recep Tayyip Erdogan called the demand "disrespectful" and said Turkey would not seek permission from others when making its defence cooperation agreements.
"Demanding that Turkey pull its soldiers is unfortunately also disrespectful toward Turkey," he said.
He said Turkey would continue to support Qatar against the many sanctions it has faced since several Arab countries moved earlier this month to isolate the country for its alleged support of terrorism.
Erdogan said he made a similar offer to Saudi Arabia to set up a base there in the past but did not hear back from the king.
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Doha received a 13-point list from Saudi Arabia, United Arab Emirates, Egypt and Bahrain that included demands to shut down the media network Al-Jazeera and cut ties with Islamist groups including the Muslim Brotherhood. The energy-rich country said it was reviewing the ultimatum but added it would not negotiate under siege.
"Here we see an attack against a state's sovereignty rights," Erdogan said.
Erdogan called the demand that Qatar shut down Al-Jazeera an attempt to take away the network's press freedom and urged rights groups to speak out against that.
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