Lanzini has endured a stop-start season with the Hammers due to injury, but the playmaker still managed five goals and six assists from 27 outings in the Premier League.
The Argentine - who has just over two years left on his contract - has previously suggested that he is happy at the London Stadium but according to The Mirror, offers may come the player's way during the summer.
Liverpool have money to spend after the January sale of Philippe Coutinho, and the signing of Lanzini would be considered close to a like-for-like replacement for the Brazilian.
West Ham would likely demand a large fee for Lanzini, but the 25-year-old's contract situation may leave the Hammers wary of holding out another 12 months.
The East London outfit could move to tie down one of their star men to a fresh deal before the start of the next campaign.