Speed, who took over following the departure last weekend of Kevin Blackwell, has remained in touch with Middlesbrough manager Gordon Strachan and his assistant Gary McAllister since the trio were players at Leeds United in the early 1990s.
However, fate has now put Speed on a collision course with the two former Scottish internationals as he strives to make an instant impact in management with the Blades.
"It's always nice to see Gary and Gordon and I am looking forward to seeing them after Sunday's game for a drink," said Speed.
"It is ironic that they are both at Middlesbrough and this would have been a difficult game at any time of the season never mind my first game in charge."
Speed, 40, was only 19 and his playing career in its infancy when Strachan, then 32, arrived at Leeds from Manchester United in March, 1989.
McAllister followed 16 months later following the club's promotion to the top flight and the trio, assisted by David Batty, formed a midfield unit that helped Leeds win the title.
Looking back on those days at Elland Road, Speed said: "Whenever people talk about that old Leeds team they immediately say what a good midfield it was. But I always say that we only won the league title because we were a good team. The midfield didn't win the league, it was the team.
"Gordon had a massive influence on my career – as much as anyone, certainly as a player," he added. "Maybe I was not as good as him, technically, but he taught me a lot and I was lucky to be a 19-year-old when he came to the club."
Speed believes Boro justify the position of bookies' favourites to win this season's Championship having invested heavily in players including Kris Boyd, and, in the last 48 hours, former Blades and Leeds defender Matthew Kilgallon.
Both sides are looking for a first league victory of the season and are under pressure to climb the table. Speed has made it clear that promotion is the target for the Blades.
The arrival of coach John Carver yesterday was the first major step in Speed's restructuring programme. Carver was on the backroom staff at Newcastle United when Speed was a player at St James' Park and had also worked at Leeds under Blackwell.
He has been allowed to leave his post as assistant to Peter Reid at Plymouth Argyle without any demand for compensation and took part in his first training session with the Blades yesterday morning.
"He has coached around the world and his experience is second to none," said Speed. "His appointment was a
no-brainer really. You need a different voice on the training field and John is a good coach and an affable, likeable fella."
Speed is still in talks with defender Jean Calve about a move to Bramall Lane from French club Nancy but is unlikely to have new options at his disposal before tomorrow's game.
Defenders Stephen Jordan and Nyron Nosworthy are still not over the injuries which have delayed their inclusion in the side but Speed may have other changes up his sleeve in the wake of last weekend's dismal
3-0 home defeat against Queens Park Rangers.
"I do think we need a few new players in before the transfer window closes," he said. "Hopefully, I will be able to do that next week. But I am fortunate to have a great bunch of professionals and I know they will give me everything I want in terms of attitude and work-rate."
Blades captain Chris Morgan is not expecting wholesale changes from the new manager at this stage.
"I don't think there will be sweeping changes because Gary agreed with a lot of things that Kevin Blackwell did and admired a lot of things, too," said Morgan. "But he is a manager in his own right now and has tried to make a couple of subtle changes over the last few days.
"He doesn't want players going into their shells, he wants us to speak to him still. He also wants people to be a bit more relaxed, especially in possession of the ball.
"We were all bitterly disappointed that Kevin Blackwell lost his job but football doesn't allow you to stand still for too long," he added.
"That decision has been made and Gary is a good appointment. I am sure Kevin will bounce back."
Team news: Former Blades defender Matt Kilgallon is set to make his debut for Boro after moving on loan from Sunderland. Captain Gary O'Neil is struggling with illness and defender Seb Hines ruled out with an ankle injury. Defenders Stephen Jordan and Nyron Nosworthy are still unavailable for the Blades while Darius Henderson, Ryan France and Lee Williamson are long-term absentees.
Last six games: Middlesbrough LDLLWD, Sheff United DWWDLL.
Last time: Middlesbrough 0 Sheff United 0; August 7, 2009; Championship.
Referee: D Whitestone (Northamptonshire).
SkyBet odds: Middlesbrough to win evens, Sheff United to win 3-1, Draw 11-5.