"Phil is a good pro and has come here to perform," Robson told the official Sheffield United website.
"I am confident he will do a good job for us and we are always looking to improve the squad. He also has experience out on-loan after spells at Burnley, Rangers and Aston Villa."
Bardsley has found first team openings limited at Old Trafford, but has already caught his new manager's eye in his short time training with the Blades.
Robson said: "He's fitted in with the lads straight away and he goes straight into contention for Saturday.
"It's important for me that Phil brings the right balance to the back four. He is a specialised full-back and that creates a more defensive sound unit."