The bronze statue was chopped off just above the ankles, with the main "body" left lying on the pavement.
It is the second time the statue, which was unveiled in June 2016, has been targeted by vandals. Police said the motive was unclear.
Messi recently saved Argentina's 2018 World Cup place when he scored a hat-trick against Ecuador in October.
Argentine media tweeted pictures of the vandalised statue.