"The power in the country has been seized in its entirety," the military group read out on NTV, CNN Turk reports.
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A Turkish presidential source told Reuters that the statement was not authorised by the military command.
Gunfire was also heard outside Istanbul police headquarters and tanks are said to be stationed outside Istanbul airport with all flights cancelled.
"We are working on the possibility of an attempt. We will not allow this attempt. Those who are in this illegal act will pay the highest price.
"There was an illegal act by a group within the military that was acting out of the chain of military command. Our people should know that we will not allow any activity that would harm democracy," Prime Minister Binali Yildirim said.
Yildirim earlier denounced the "illegal action by a military group", stressing it was not a coup. He maintained that the government remained in charge.