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.
In 2011 Anzhi signed former Barça striker Samuel Eto'o, making him the highest-paid player in world football.
El Mundo Deportivo said Messi rejected the Russian move because of his desire to continue winning titles at Barcelona.
He scored a record 91 goals in 2012 and is expected to win his fourth consecutive Ballon d'Or.
Neither is Messi short of cash - in March it was revealed he earns £27m per year, topping up his £13m salary with sponsorships and endorsements.
Every player in Spanish football must have a minimum fee release clause in their contract. Cristiano Ronaldo's is the highest, standing at €1bn (£800m)