The 22-year old former Adelaide United attacker was sent out on loan to Beta Ethniki (second division) club Kerkyra last term and was believed to be on trial with AEK during last week's Festival of Football in Sydney.
According to coach Dusan Bajevic, Burns performed well enough to be considered for the upcoming Greek season but admitted a final decision would not be made until the team returned home.
''We decided last year to send him to another team for more experience, but I know well he has good qualities and he showed us in this tournament the best way,'' Bajevic said.
''I believe with all that he has showed to us, he will help us very much this season. I believe he will stay, but I should discuss this when I return to Athens. When he came to AEK he wasn't ready to be a starting player, but now he is. He is not an unknown player to me and he has improved very much.''
Burns appeared in all three of AEK's matches, having a hand in two goals against Blackburn Rovers on Wednesday before scoring the winner against Rangers on Saturday as his side went through undefeated to claim the tournament.