Here, the PA news agency looks at possible replacements for the 53-year-old.
The Northern Irishman has emerged as one of the shock early favourites with the bookmakers to succeed Wilder. Lennon left Celtic last month after overseeing a disastrous campaign in which the club were attempting to win 10 Scottish titles in a row. Glasgow rivals Rangers have since been crowned champions and are 20 points clear at the top of the table. He did help Celtic win five pieces of silverware, however, during his second spell at Parkhead and this week said he was keen for a quick return to management.
The 43-year-old is actually one of the favourites to succeed Lennon at Celtic. Long regarded as one of the brightest young managerial prospects in English football, Howe is available after leaving Bournemouth last summer. Much like Wilder at Bramall Lane, he led the Cherries through the divisions to the Premier League – and then oversaw a five-year stay in the top flight. His reputation remains intact despite overseeing Bournemouth's relegation last season.
Has been out of work since he was sacked by Chelsea in January after 18 months in charge, a decision which seems justified given the progress the team have made since Thomas Tuchel's appointment. Lampard did much good during his time in charge of Chelsea, bringing through the club's youngsters and establishing them in the first team. Although he was criticised towards the end for failing to get the best out of big-money signings like Timo Werner and Kai Havertz. The 42-year-old's first job was in the Championship with Derby, getting the Rams to the play-off final.
Lampard's former England and Chelsea team-mate is expected to step up to a top job sooner or later after two years working as assistant boss to Dean Smith at Aston Villa. The 40-year-old, who won multiple honours during his illustrious playing career, has also been linked with other posts at Championship clubs in the past, including Derby, Bristol City and Bournemouth.
The Blades' Under-23 coach has been installed as the early favourite with the bookmakers, and he will be in charge of the first team at the King Power Stadium against Leicester on Sunday. On that basis, you could argue it is his job to lose if he can impress the owners during the club's remaining Premier League fixtures. The 43-year-old has had stints as manager of Barnsley and Leeds and was sacked from his most recent job with Hibernian, but he is well-regarded at Bramall Lane and it would be an easy transition for him to step up to the first-team set-up.