Blackwell was sacked as Blades boss on Saturday following a 3-0 home loss to Queens Park Rangers.
The defeat followed last weekend's season-opening 1-1 Championship draw with Cardiff City and a 2-0 midweek League Cup exit at the hands of League One Hartlepool United.
Speed, 40, who is a coach at the club, was always favourite to succeed him. It will be his first management role.
He was wanted by Swansea City last month when they were looking to replace Paulo Sousa, but the Welsh club were not given permission to talk to him.
Speed has been at Bramall Lane since early 2008. He retired from playing at the end of last season but was named as a substitute for the reverse at Hartlepool just under a week ago.
The club have not officially confirmed the appointment, but are expected to do so on Tuesday when he will be unveiled at a press conference.
Harry Redknapp, the Tottenham Hotspur manager, has suggested Craig Bellamy would be wasting his talent if he joins Cardiff City on loan.
The Wales international forward looks poised to leave Manchester City before the end of the transfer window, but his club are reluctant to let him join a club they see as rivals for a Champions League place this season.
That has scuppered Redknapp's attempts to take him to White Hart Lane. Cardiff hold a strong pull for Bellamy, who has said he wants to play for the Bluebirds before ending his career.
"Good luck to Cardiff but it would be a waste of an outstanding player, he should stay in the Premier League for sure," said Redknapp.
"He's a Premier League player - a top Premier League player in my opinion."
Middlesbrough have agreed to sell goalkeeper Brad Jones to Liverpool for a fee of £2.3million.
Jones, 28, had a medical at Anfield on Monday and is expected to complete his move on Tuesday.
Kevin Thompson, the Boro winger, will miss the next two months after breaking a leg during Saturday's goalless draw with Leicester City.
The Scottish international, 26, broke the fibula in his right leg during only his second appearance since joining Boro from the Scottish champions Rangers this summer.