The Tottenham defender is currently undergoing tests at the club's training ground and will talk terms with the Baggies
The club are confident the deal will be done on Friday and the 28-year-old could be available to face Chelsea in the Barclays Premier League on Sunday if the deal is completed in time.
Boss Tony Pulis said: "The lad is here and having a medical and we're hoping to get it done.
"He's a very good defender coming from a top club in Tottenham and the boy wants to play regular football.
"He wants to show that he can not only compete but do well in the Premier League and I think that ambition, drive and desire is what you need in your club.
"He's been a top player for Sevilla and I think he will be a top signing for the club."
Pulis also denied the deal is linked with Tottenham's pursuit of Saido Berahino after the Baggies rejected Spurs' bid for the striker this week.
He added: "That deal has nothing to do with Saido's position at the football club
We want Saido to stay
You never say never but we want him to stay
It would be in his best interests."
Fazio signed a four-year contract at Tottenham last summer after joining from Sevilla, where he won the Europa League in 2014.
He made 31 appearances for Spurs last season but is yet to play this term after the signing of Toby Alderweireld.
Fazio's imminent arrival casts doubt over the future of Joleon Lescott at The Hawthorns, with Pulis refusing to offer guarantees over his future.
Source : PA
Source: PA