Speaking exclusively to Football Insider, the former Scotland boss suggested Ange Postecoglou needs to sit down and talk to the 26-year-old to find out how to get the best out of him.
Bolingoli, who was cast out at Parkhead last season, made his second start of the season in the win over Motherwell on Saturday.
Postecoglou has left Bolingoli out of the matchday squad for every other of the Hoops’ Premiership games.
Celtic have also not registered Bolingoli for their Europa League squad.
However, McLeish insists the left-back has the ability to become a first-team mainstay despite crashing down the Parkhead pecking order.
He suggested Postecoglou would have to use his man-management skills to figure out the best way to get the defender firing on all cylinders.
Asked he thought Bolingoli could truly deliver his best performances coming in and out of the side, McLeish told Football Insider’s Russell Edge: “I guess it depends on getting close to your players at club level.
“Feeling what ticks the boxes for them. Sometimes it can be: ‘I need to play every week to get to my best level.’
“Or is he the type of player who comes in and says: ‘I’ll show you gaffer. When you pick me today, I’m going to be one of the top performers.’
“There’s all these kind of motivational factors. You would think Postecoglou seems like a guy who would happily talk to his players and help the mental side.
“We’ll see with Bolingoli. He’s definitely got the quality to do it, but can he do it for 10 games in a row? Can he make himself an absolute first pick? That’s his challenge.”