Laurent, 24, has been training with the Blades after spending the last two seasons with Southend United.
Speed, who signed defender Jean Calve on loan from French club Nancy earlier this week, admits that he still wants to strengthen his squad before the transfer window closes on Tuesday.
“I hope we can do business and bring at least one more in by the time the window shuts,” he said. “We have loads of lists and names kicking around – too many to remember.”
The Blades will have both Calve and Stephen Jordan available for the first time tomorrow against Preston North End. Jordan, who was former manager Kevin Blackwell’s final signing following his release by Burnley, has missed the last three games with a calf problem.
Defender Jonathan Fortune, who was released by the Blades this summer, is on the brink of securing a deal with his former club Charlton Athletic.