Lanzini still has two years left on his current £35,000-a-week Hammers deal and the club have the option to extend his terms by a further year.
Indeed, Metro reports that the Argentina global is "firmly" on the Reds' radar ahead of the summer transfer window, with the Merseyside outfit believed to be on the lookout for a replacement for Philippe Coutinho at Anfield.
Under contract until 2020, there have been reports that West Ham will be offering Lanzini a new and improved contract but David Moyes has confirmed that the club will not be opening talks until their Premier League status is secured.
Lanzini has continued to impress for the Hammers this season, scoring three goals and providing five assists despite missing ten Premier League fixtures through injury.
At a press conference, the Scot told reporters: "You always want to keep your best players but there's no truth in that yet". We won't look at that [extending Lanzini's contract] until we know what league we are in.
Quick start can help us beat Swansea - West Ham star Lanzini
Moyes has warned his team to expect a hard test when they travel to face a resurgent Swansea in tomrrow's Premier League clash.
With ten games of the season remaining, West Ham are now 13th in the top-flight, three points above the relegation places.
He also insists that a point would not be a bad thing for either team, with both battling relegation, and boldly states that any one of eight sides could be the side who struggles to pick up victories between now and the end of the season, resulting in the drop.
"It is a tough game going there but we have had some good results away from home, hopefully we can continue that". "There are no easy games, but I feel we can do it and that it should be six points".
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