The Argentina international broke Muller's 40-year record of scoring the most goals in a calendar year by nabbing his 86th goal in 2012 for the Spanish giants.
The 25-year-old sent the ex-Bayern Munich frontman his famous number 10 shirt, with the message, "For Gerd Muller, with my respect and admiration. A hug. Leo," written on the back after reaching the feat.
Muller has claimed that he is "happy and proud" to be given the shirt and described Messi as "the best player in the world", according to the Daily Mail.
Meanwhile, Messi has added to his trophy cabinet by claiming his fourth Ballon d'Or honour.