Quinton Fortune, who played alongside former United captain Keane, believes the Irishman would raise standards at Old Trafford, with current manager Michael Carrick facing growing pressure following a poor run of results.
United are without a win in seven matches across all competitions after Wednesday's 4-1 Champions League defeat by Como, which has intensified scrutiny of Carrick's position.
The 45-year-old is expected to remain in charge for Sunday's Premier League match against Bournemouth, but the result could prove significant for his future.
Fortune believes the former midfielder's leadership and demands for excellence could have a positive impact if he returned to the coaching staff.
"Roy Keane coaching at Man Utd would be a lot of fun," Fortune told BOYLE Sports, who offer the latest Premier League odds.
"I was fortunate to play alongside Roy, and he was an unbelievable leader. He pushed every team to success and raised everyone's standards.
"I thoroughly enjoyed working with him in training and matches because he demanded excellence and made you a better player."
Keane is known for his forthright criticism of United's performances, but Fortune believes his record as a player gives weight to his opinions.
"I have the utmost respect for Roy. If he speaks out, it's because he wants excellence," he added.
"People talk about walking the walk and talking the talk, Roy did both at the highest level on the pitch. Having him around at Man Utd would shake things up and be a lot of fun."
Fortune also believes Man Utd should pursue Harry Kane if there is any possibility of bringing the England captain to Old Trafford.
Kane, 33, is currently the bookmakers' favourite to win the 2026 Ballon d'Or, with the ceremony due to take place in London on Monday.
"He's not young but given what he's achieved, Man Utd should go and get Harry Kane as soon as possible, whatever it takes," Fortune said.
"He is exactly what they need.
"If there's any chance of bringing Harry Kane in, do whatever it takes because the man is a natural-born goalscorer," he said.
"Give him service and he will score goals. It's as simple as that. You give him service, he will put the ball in the back of the net.
"Just go and get him. Simple as that. If there's a possibility, there's a small chance of going to it. Don't wait any longer because what a player he is, my goodness."