He has been offered the position by chairman Kevin McCabe following the departure of Kevin Blackwell on Saturday.
Speed, already on the coaching staff at Bramall Lane, was blocked from speaking to Swansea City about their then vacant manager's job during the summer.
It is understood the refusal was based on the Blades board eyeing Speed as a potential successor should Blackwell leave the club once the new season was underway.
Former Leeds and Luton manager Blackwell, who had worked as assistant to Neil Warnock at various clubs for 12 years, three of them at Bramall Lane before taking up a senior coaching role at Elland Road in 2003, still had two years left on his contract.
The 51-year-old was installed as Blades manager in February 2008 following the sacking of Bryan Robson, but after leading the Blades to the play-off final in 2009 - they lost out to Burnley - he has found himself under increasing pressure after his failure to mount a promotion challenge last season.
Former Leeds, Everton, Newcastle and Bolton midfielder Speed, who held the record of most Barclays Premier League appearances, 535, before being overtaken by David James and Ryan Giggs, joined the Blades initially as a player for 250,000 from Bolton on January 1, 2008.
Speed made 38 appearances for United, but his playing days were curtailed by a persistent back injury and he focused on his coaching duties towards the end of 2009.
Swansea were keen for Speed to replace Paulo Sousa as manager earlier this summer, but the Welsh club were refused permission to speak to him by the Blades, for whom the former Wales international is still registered as a player.