Keane claims that the Scottish giants can't be called a big club anymore if they fail to improve their results in Europe.
He told ITV: "I know it's difficult after a game, and if you have an off night in Barcelona you are in big trouble.
"But, on one hand we are being told that Celtic are a great club, great traditions, great history, and they go and lose 7-0 to Barcelona and people almost say, 'Well these things happen, it's acceptable'.
"He spoke about the game the previous week against Rangers. They won 5-1 against one of the worst Rangers teams I have seen in a long time, who finished with 10 men for the last 15-20 minutes. So that idea they were tired as well.
"All I hear sometimes when people talk about Celtic is about winning the league, but I think they need to do a lot better in Europe. It's easier said than done of course and this time round it's obviously difficult with the group they are in.
"But, even in the qualifying games, they were poor. They scraped through and didn't win any of their away games.
"I think the whole mindset has to change and don't accept, particularly away from home, being the whipping boys of Europe."