The 31-year-old never established himself as a regular first-team starter after joining from Atletico Madrid in January 2010 but he still managed some stand-out performances.
Rodriguez scored 17 goals in 73 appearances, the highlights being two hat-tricks in the space of a couple of weeks against Birmingham and Fulham in 2011, while last season he scored in two victories over Chelsea in quick succession at Stamford Bridge.
The Argentinian, a fans' favourite if not one 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."
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