BBC Radio Sheffield says the 38-year-old former Wales international is expected to join on loan before making a permanent move.
Speed could even sign in time to be in the Blades' squad for their New Year's Day clash at Wolves.
He was made player-coach by former Bolton boss Sammy Lee, but stepped down and is now ready to quit The Reebok.
Speed admits he may need to move to get more football in the closing stages of his career.
He said: "I'm not in the team at Bolton. I've no complaints about that, the team are playing well but I need to be playing games.
"I will see what happens. I am fit enough to play. I have not got much time left and I want to play football."