Spanish sports daily El Mundo Deportivo reports that the club agreed to pay more than double the world record transfer fee, and offered the Argentine a €30m (£24.6m) annual pay packet after tax.
Once that sum was offered Barcelona were contractually obliged to accept it - but from that point it was up to the player whether he was interested in leaving the club.
The bid came during Messi's recent contract negotiations with Barcelona.
However, the 25-year-old rejected the move and signed a two-year extension with Barça that ties him to the Camp Nou until 2018.
Although the club was not named it is likely to have been Anzhi Makachkala, who are owned by billionaire Suleyman Kerimov.