The Hammers are eager to fend off competition for the Argentine from Turkish side Fenerbahce.
The playmaker missed most of the past season because of a knee injury suffered ahead of last summer's World Cup. Lanzini made just 10 Premier League appearances for the Hammers, eight of them starts, and scored one goal after finally making his comeback in February.
The 26-year-old's current deal runs until June 2020, and the east London club also have a two-year option. But boss Manuel Pellegrini is a big fan of Lanzini, who is among the players key to his plans as he looks to build on their 10th place finish from his first season in charge.
The move will undoubtedly bring a sigh of relief for Hammers fans', as the club already look resigned to losing Marko Arnautovic for the new season.