Burns moves to AEK
Adelaide chief executive Sam Ciccarello said Burns, 20, had been looking for overseas opportunities for some time.
"We respect and support Nathan's desire to compete in a European league and he most certainly goes with our very best wishes," Ciccarello said in a statement. "It has been wonderful for the club to have brought Nathan to Adelaide and to have been a key to his professional personal development over the past couple of years."
He predicted a big future for Burns at AEK, who came second in the Greek league last season.
Burns scored 11 goals in 54 appearances for Adelaide and earned a cap for Australia's senior national side last year when he came on as a second-half substitute against Singapore.