Sam Allardyce announced on Wednesday night that he would not be staying on as boss at the Hawthorns club, who have been relegated ahead of the final game of the Premier League season.
Stocksbridge-born Wilder left the Blades in March after a difficult second campaign in the top-flight, with the Blades officially relegated a month later following a 1-0 defeat at Wolverhampton Wanderers.
Wilder took over at United in 2016 and earned promotion from League One with a club-record 100 points. He then helped the club earn a place in the Premier League in the 2018-19 season and secured a ninth-place finish in their first campaign back in the top flight.
The Blades were bottom of the table when he left the club earlier this year, with 14 points from 28 games.
Reports have claimed that the 53-year-old is one of West Brom's top candidates to fill the head coach vacancy, along with Frank Lampard and Michael Appleton.
The Baggies play their final Premier League match of the season at Elland Road tomorrow, with Leeds United preparing to welcome fans into their home stadium for the first time this season.
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