The Argentina forward is out of contract in the summer.
The likelihood of another early Champions League exit for Barcelona has fuelled speculation he is could quit the Nou Camp.
Uncertainty remains due to the ongoing presidential elections at Barca, with all candidates saying they are determined to keep the 33-year-old.
Messi has won six Ballon D'Or trophies in a stellar career that has seen him generally accepted as one of the finest players ever.
It has been reported City have tabled a revised offer for Messi but BBC Sport reports a club source says this is not true - and that no offers have been made, either in the summer, or since and there are currently no talks.