The Londoners want to renew Lanzini's contract as soon as possible because his contract expires in June 2020 and from January, the Argentine will be free to negotiate with any other club.
West Ham bought him in 2016 for 12 million euros from Al-Jazira after having him on loan previously.
This term, Lanzini has started both Premier League games and has played all 180 minutes. Pellegrini has asked the board to make an effort to renew his contract and to not let time pass.
Lanzini is apparently happy at the club, but will demand a pay rise in order for West Ham to activate the option to extend his contract until June 2022.