The club are reportedly keen to keep hold of the Uruguayan, who has been linked with a move away this summer.
Hernandez could become Hull's most expensive player ever
Several Italian clubs are also thought to be monitoring the situation, although the Tigers have now reportedly offered him a deal that would make him the club's highest-paid player ever.
The Hull Daily Mail says the contract has already been offered and discussions with the 26-year-old's agent have begun.
Despite reports suggesting he is keen for a move away from the club, Hernandez is understood to open to the possibility of extending his stay with City and in the Premier League.
While speculation surrounding his future has been rife this summer, there is reportedly no buy-out clause in his contract, while any new deal for the player will not rely on a takeover of the club.
Hull are hopeful that a deal can be agreed in the next week, with Hernandez heralded as one of their most successful signings in recent years following his £10 million move from Palermo in 2014.