Brendan Rodgers’ side conceded a stoppage-time winner to lose 2-1 to Lazio in their Champions League group game at Parkhead on Wednesday night.
Celtic have lost their opening two group games, against Feyenoord and Lazio, respectively, and are sitting at the bottom of Group E of the Champions League.
Bonner admitted that there were expectations of Celtic ahead of the Champions League this season and fans were looking for some tangible positives in Europe.
However, the former Celtic star insisted that the team have not looked at their best since Rodgers came in at the start of the campaign.
He stressed that they have to finish third in their group as Celtic need some signs to suggest that they have made some progress in Europe.
Bonner said on BBC Sportsound: “We were all looking forward to the Champions League for Celtic this season.
“We thought they had something about them, that they could go and actually compete.
“Then, the change in management, then they went off the boil a little bit.
“Third place, at least, is absolutely vital for Celtic.
“They have to show progression.”
For the moment, Celtic are facing an uphill battle to even finish third in their Champions League group this season.
