The Argentine playmaker is a key member of Manuel Pellegrini's expansive side.
The 26-year-old could be about to commit the key years of his career to the London Stadium club.
His current deal has a year to run, but he is in talks to extend his stay to 2022 at the minimum.
The Hammers do have a two-year option on his existing deal, but they would prefer to reward the player with a new contract.
Lanzini has welcomed discussions, telling Sky Sports: "Yes, we are talking with the club about staying and, of course, I want to stay.
"It is like my home. It is my fifth season now, I love the fans, the fans love me and the club is amazing."