On the other hand, West Ham has confirmed the signing of striker Perez after reaching an agreement with England's Arsenal.
The Mexico worldwide scored just eight goals in 28 league appearances last season but is looking forward to working with Pellegrini.
Sanchez joined the Viola on loan for the 2016-17 campaign, making 41 appearances in that time, but made just nine in Serie A last season and was loaned to Espanyol for the final six months.
"I'm very happy to be here at such an historic club", he told the club's official website.
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"I always dreamed of playing in the Premier League", Sanchez said.
Former Manchester City boss Pellegrini's spending since replacing David Moyes has reached around 100 million pounds as the Chilean tries to turn West Ham into a side capable of challenging for Europe rather than battling to avoid relegation.
"He has great technical quality, and can play as both the main striker or the second striker in a deeper role".
Perez managed to score eight goals in 35 matches for Deportivo who have been relegated into the Segunda division in Spain last season.