The 40-year-old was offered the job in the wake of Kevin Blackwell's departure and is understood to have agreed terms with Kevin McCabe yesterday after flying out to meet the Blades chairman in Spain.
Speed, immediately installed as the favourite to take charge following the dramatic events of last Saturday, was the board's preferred choice amid reported interest from several high-profile candidates.
He has been working as first-team coach at Bramall Lane for the past couple of years but has been keen to make the step up.
The former Wales international was wanted by Hull City during the summer only for Nigel Pearson to suddenly become available which saw the Tigers' board switch their attentions to the then Leicester City chief.
Swansea also asked for permission to speak to Speed over a vacancy that had been created by Paulo Sousa's departure to the Walkers Stadium. He had been in the running to take over at the Liberty Stadium a year earlier only to miss out to Sousa.
The Blades, who have called a press conference for 2.30pm today, turned down this summer's request from the Swans – a move that suggested the directors saw the former Leeds and Newcastle midfielder as a potential next appointment should the season not pan out as hoped under Blackwell.
Now in full charge, Speed's first task after meeting the media will be to bolster his backroom staff following the departure yesterday of Sam Ellis, Blackwell's right-hand man. It is believed both the Blades and their new manager favour an appointment who boasts plenty of coaching experience.
The Yorkshire Post understands Ricky Sbragia would have been Speed's preferred choice as No 2 if he had taken over at Hull's KC Stadium during the summer, suggesting the former Sunderland manager may be an option at the Lane.
The new Blades manager, who completed his UEFA Pro-Licence qualification last year, has 10 months remaining on the final year of his contract as a player-coach. His bow in charge will come on Sunday at Middlesbrough.