Speed, who only took the reins at Bramall Lane in August following the dismissal of Kevin Blackwell, has been named as John Toshack`s successor, leaving United looking for their third manager of the season.
John Carver will take temporary charge of the Blades while they select their new number one.
"We didn`t want to lose him but we have to respect Gary`s decision given the uniqueness of the opportunity," chief executive Trevor Birch told the club`s official website.
"This is a prestigious appointment and we wish him every success and thank him for his contribution to the club over the last three years.
"The club is very disappointed to lose Gary but will continue to move forward. We have started our search for a permanent replacement and are intent on getting the right man who will be aligned with our strategic aims in terms of playing style, development and recruitment, working closely with Academy manager John Pemberton and myself."
Source: PA
Source: PA