The Argentinian, popular with supporters if not with the three different managers he had at Anfield, wrote an open letter to supporters.
"Dear Reds, I am leaving Liverpool FC today," it said. "Before signing for LFC, I just saw the club as one of the greatest institutions in football. After my time here I can confirm that this is not just a great club but also a great family.
"I have tried to give everything every day that I was wearing the LFC crest. It has been a great honour to defend this shirt during two and a half years.
"I am returning home with a suitcase full of great memories, good friends within the team and also within this great family that works every day in the club. Thanks a lot for your support. Hasta la vista."
A statement from the club read: "Everyone at Liverpool FC would like to thank Maxi for his contribution to the club and wish him all the best for the future."