The Swede's contract expired at the end of May and there was debate whether he would stay on at Parkhead for at least another season.
However, it is understood that Lustig and Celtic were far apart in contract talks meaning they were never able to come to an agreement.
The 32-year-old was keen for a three-year-deal but the Scottish champions, taking his age into consideration, were only prepared to offer him a 12-month extension, according to the Daily Record.
Lustig is believed to be nearing a deal with Belgian side KAA Gent with Malmo in his homeland also interested.
The player previously stated: "I am proud of how I have handled the situation and those closest to me, the players and the leaders at Celtic know what has happened. It could have been handled differently.”
And he uploaded a heartfelt message on Instagram confirming his exit and paying tribute to the club and supporters.
"I had no idea how much this club and everything that comes with it would mean to me when I signed 7.5 years ago. It's time for me and my family to move on, start a new chapter in life.
"To make a post on Instagram isn't enough to express my feelings and thoughts about the situation. I wish I would've had the chance to say a proper good bye.
"I just want to say thank you to everyone I've had the honour to meet and work with at Celtic, all the players and staff. The memories will stay with me forever.
"Thank you to all of you that made Glasgow feel like home for me and my family for so many years. I will never forget your support.
"Hopefully I'll see you all soon. I'll definitely be back for the 10 in a row party!"
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Both Neil Lennon and club chief Peter Lawwell paid tribute to the Swede who was at he club for seven-and-a-half years, winning 16 trophies.
The manager said: "“I was delighted to bring Mika to Celtic and he has been phenomenal for me and the club.
“Mika has chosen to move elsewhere but that is absolutely his right. These things happen but it will not in any way take away from what he achieved at Celtic.
“I wish Mika nothing but success for the future. He has been a brilliant servant to the club and has made a massive contribution to everything we have achieved across so many years. It has been a privilege to work with Mika.
“A top international player, he always gave everything for the team. He gave his heart and soul to every performance and I loved having him here.”
Lawwell added: “For Celtic he has been a real force, a highly-talented player who always fought for the cause. He has been crucial to so much of what we’ve achieved.
“Everyone at the club sends their best wishes to Mika and his family.”