One national newspaper claimed Wilder had been approached about coming back to the club he led to ninth in the 2019-20 Premier League, only to leave the club midway through the following season, when they were relegated from the Premier League.
Wilder was replaced on a caretaker basis by Paul Heckingbottom, who was given the job permanently in November 2022 after initially returning to his job with the club's under-21s.
Heckingbottom has had a difficult start to this season, exacerbated by Saturday's 2-1 defeat at Tottenham Hotspur, which means they have taken one point from their opening five Premier League matches, as well as being knocked out of the League Cup.
But the club are aware of the difficult task he has been handed, given by Premier League standards a negligible budget to upgrade his squad after winning promotion from the Championship until the sale of key players Iliman Ndiaye and Sander Berge raised extra funds.
Bettis, who runs the club on behalf of owner Prince Abdullah was unequivocal when asked in his annual briefing with local media about Heckingbottom's future in the first week of the transfer window, insisting the Blades would stand by their man.
Since leaving Bramall Lane, Wilder has had short and difficult spells in charge of Middlesbrough and Watford.
If Bettis is true to his word, then the former Blades defender returning for a second spell as manager of the club he supported as a boy is not on the agenda any time soon.