United have won nine, drawn two and lost two of their 13 matches since Carrick's arrival as he took over from former manager Ruben Amorim.
The former midfielder is on the verge of leading the Red Devils back to the Champions League as the board weigh up whether he should be handed the permanent role.
Under Carrick, United are 11 points clear of sixth place in the Premier League with 12 points left to play for and speaking on The Overlap and Betfair's Stick to Cricket show, Bruce believes he should be given a chance.
“I sincerely hope Michael Carrick gets the Manchester United job. Why not? He's certainly put himself in pole position. He gets the club; he understands the club because he's been there and played for it. So, I hope he does get it permanently next season. Why not?”
Bruce, who had been managing League One outfit Blackpool before he was sacked by the club in late October, also spoke on his next steps as he revealed his time in the game may be over.
“I can never say never, but I honestly believe that I've had my time in management now. I thoroughly enjoyed it and still love it to this day, but I think there comes a time when you have to call it.
“You never say never; maybe something like an international job. I had a conversation with Ireland a few years ago, and would you believe, before Steve McClaren, I had a chat with the Jamaican FA. They went for Steve McClaren in the end.”
Bruce is just 65 years old and the return of Roy Hodgson to Bristol City at 78 shows that he may yet return to management if given the chance.