The Argentine defender was one of the stars of his side’s Europa League success last season, helping the La Liga outfit to overhaul Benfica in the final.
His performances appear to have caught the eyes of Brendan Rodgers and Arsene Wenger, which could spark a bidding war for the 27-year-old.
Liverpool are known to be in market for enforcements at the back after their poor defensive record ultimately cost them a first league title in 24 years, while Arsenal are also looking for additions as captain Thomas Vermaelen edges towards the exit door.
Sevilla are desperate to hold onto their star, but bids of over £8m may be too tempting to pass on with financial difficulties in Spanish football.
Fazio has been offered a new deal, yet he is understood to be aware that more cash awaits him in the Premier League.
Barcelona are also in the hunt as they plot their summer overhaul on the back of a dismal showing last season.