Fiorentina are looking to offload several players this summer in order to trim their squad and the Colombia worldwide is believed to be on the cull list.
Capped 88 times by Colombia, Sanchez would add experience to a position where West Ham do not have depth.
If West Ham complete the deal before Thursday's transfer deadline day then the 29-year-old will become the club's ninth signing of the summer and will take the total spending close to £100m.
With further midfield signings on the agenda, West Ham could be tempted to cash in.
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Perez previously played with Wilshere at Arsenal and the duo look set to reunite at the London Stadium for the new season.
And new boss Pellegrini is looking to make former Aston Villa midfielder Carlos Sanchez his 10th - and final - capture of the window.
Perez, who has signed a three-year deal, spent just two years at the Emirates Stadium - one of those out on loan at Deportivo - and scored seven goals in 21 appearances.
Sanchez spent second half of last season on loan at Espanyol and his contract with Fiorentina expires at the end of next season.