Lanzini, 24, emerged as a top young star with River Plate in his native Argentina, but moved on to Al Jazira in 2014, spending just a year in the United Arab Emirates before joining West Ham on loan in 2015.
Lanzini has stepped up and taken over creative duties since Dimitri Payet's departure in January, and he scored in the 4-1 defeat to Liverpool last weekend, which proved to be Slaven Bilic's final game in charge.
Tottenham may be looking for another creative player to take some pressure of Christian Eriksen, though the possibility of a bid for Lanzini may lead some fans to believe that Lanzini may be lined up as a replacement for Eriksen.
West Ham would loathe to lose their star man to a London rival, but with Lanzini allegedly earning just £35,000-a-week, a move to a Champions League club contending for the Premier League title could be appealing.
The report adds that Everton failed with a summer bid for Lanzini and signed Gylfi Sigurdsson instead, but Tottenham may now attempt to try their luck, despite the frayed relations between themselves and West Ham.