The Baggies are hoping to capitalise on unrest behind the scenes at Bramall Lane, where joint-owners Kevin McCabe and Prince Abdullah are embroiled in a bitter feud.
Despite that, taking Wilder away from United, the club he has supported since a boy, would represent a remarkable coup for the Baggies.
Middlesbrough, who recently parted ways with Tony Pulis, are also believed to be keen.
Having taken the Blades up to the Premier League with a bottom-half budget, Wilder was named the Championship Manager of the Year and the LMA Manager of the Year for all 92 clubs of the top four divisions.
It’s believed he’s growing frustrated with political in-fighting behind the scenes at Sheffield United – but Albion’s interest could also be used to negotiate a better contract at the Blades.
Wilder’s potential availability, however, is one of the reasons Albion’s managerial search is dragging on longer than anticipated.
The Baggies had an initial four-man shortlist when the season finished, but that has since evolved due to a number of candidates becoming available.
Former Wolfsburg boss Bruno Labbadia, who was spoken to twice earlier this week, remains in the running for the job.
It’s believed there is also another unnamed contender being considered.
Wilder was one of Albion’s preferred candidates when they sacked Darren Moore in March but United were in the middle of a promotion race then.
The Baggies reportedly planned to approach the Blades in the summer if Wilder missed out on promotion, but they still remain keen on him.
Wilder has been a success at pretty much every club he’s worked at. After taking Oxford United out of the Conference, he won promotion out of League Two with Northampton Town.
When he took over Sheffield United, they were a League One club, but two promotions in three years has taken the Blades to the top tier.
It’s been two and a half weeks since Albion’s season finished with defeat to Aston Villa in the play-offs and the players return to training in just over three weeks’ time.