The Parkhead outfit are pursuing an unprecedented eighth domestic treble this season, having already won the Scottish League Cup and the league title, and booked their berth in the Scottish Cup final.
Despite the domestic success, Celtic have struggled to make an impact on the Champions League in recent years and last progressed past the group stage in the 2012/13 season.
They are now set to jump straight into the group stage next term after winning the title, offering them another opportunity.
Bonner thinks that Postecoglou will want to make his mark in the Champions League with Celtic before leaving the club, and he believes that his former side need some reinforcements to compete in the competition given its increased difficulty.
“There will be a day when he leaves but I hope it’s a while yet”, Bonner said on BBC Sportsound.
“I think Ange Postecoglou will want to take the club to the Champions League and see how he can improve them there.
“That will take another couple of players coming in to bring them to a level where they can compete.
“You can see there is a weakness without [Cameron] Carter-Vickers.
“I think that’s an area that needs strengthened.
“Do they need a top star in the middle of the pitch?”
Celtic are expected to look to strengthen the squad over the summer transfer window and it remains to be seen if Postecoglou might again look to Japan.