Fiorentina are looking to offload several players this summer in order to trim their squad and the Colombia global is believed to be on the cull list.
West Brom have also been interested in Sanchez, but look to have been beaten to the punch by West Ham.
And new boss Pellegrini is looking to make former Aston Villa midfielder Carlos Sanchez his 10th - and final - capture of the window.
The signings of Jorginho and Mateo Kovacic have pushed the England worldwide further down the pecking order this summer and West Ham are interested in a loan deal - although the Independent states they will face competition from Leicester.
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Sanchez spent second half of last season on loan at Espanyol and his contract with Fiorentina expires at the end of next season.
The 29-year-old is due to undergo a medical with the Hammers on Tuesday. "I've loved my time here when I've been involved and playing, but obviously a year ago I went out and played games and bonded with the Aston Villa fans, staff and players and everyone around the city, so I appreciate them for that".
'West Ham have great fans and play in an unbelievable stadium, so to sign here brings me great joy.
Despite only securing his switch to the Olympic Stadium today, Perez could be a part of the West Ham squad for their opening Premier League fixture this weekend. He left for Fiorentina on loan after that relegation and made that move permanent a year ago.