Pellegrini has overhauled West Ham's squad this summer breaking the club's transfer record to sign Felipe Anderson from Lazio while signing Jack Wilshere, Fabian Balbuena, Andriy Yarmolenko, Lukasz Fabianski, Ryan Fredericks, Issa Diop and youngster Xande Silva.
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West Ham already boast Javier Hernandez and Andy Carroll up front, while Toni Martinez has impressed in pre-season and Marko Arnautovic can play up front.
Sanchez's future appears to be away from Fiorentina after spending last season on loan at La Liga side Espanyol.
"He has great technical quality, and can play as both the main striker or the second striker in a deeper role".
The striker said: 'Today I would like to thank Arsenal fans for their support, they always showed me affection and love from the first time I arrived in London. He can also cover other positions but it is as a defensive midfielder that he really excels.