According to TMW, the Hammer's have made contact about Vazquez, as they look to strengthen their squad after the injury to key player Manuel Lanzini.
The fan favourite got injured during preparations for the 2018 World Cup with Argentina, and may miss a full season of action - with the Argentine midfielder seen as a straight swap for Lanzini in the Hammers' side.
The 29-year-old's contract runs until 2021, with the midfield maestro making 60 appearances in two La Liga seasons for Sevilla, scoring 11 goals, and making nine assists after joining Sevilla in 2015 in a €15m move from then-Serie A side Palermo.
Vazquez is a versatile player, with his passing range and dribbling skills being his key assets. The Argentine can play across the midfield, as well as behind the forward.
A move to the London Stadium would see him join Jack Wilshere, whose arrival on a three year deal was confirmed on Monday afternoon after the England international left Arsenal on a free transfer.
Pellegrini has been busy in the transfer window since his arrival at the start of the summer, making three signings including Issa Diop for a club record fee from Toulouse, Lukasz Fabianki from a relegated Swansea side and Ryan Fredericks, who has joined up with the Hammers from newly promoted side Fulham.