Mr McLeish said he did not believe the clubs’ management teams have been outspoken enough after the angry scenes at Wednesday’s game.
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His comments come as it emerged police are investigating a reported 200 complaints made to them by fans, including allegations that Celtic manager Neil Lennon racially abused Rangers player El Hadji Diouf on the touchline.
Lennon called the claims “malicious and despicable”, with others claiming a whispering campaign among Rangers fans was responsible for the accusations. It is also alleged a Celtic fan taunted the Senegalese-born player with racial abuse, which is said to have been captured on broadcast footage of the game.
Strathclyde Police refused to comment on the nature of the allegations, including whether Lennon and Diouf are due to be interviewed by officers today.
Mr McLeish spoke out as Holyrood prepares to host its first Old Firm summit tomorrow. It was requested by Chief Constable Stephen House on Thursday when the impact of the game, won by Celtic 1-0, became clear.
Both clubs have said they will attend and are keen to contribute to the debate, but the former first minister, who last year led a comprehensive review of Scottish football, from grassroots to the organisation of the Scottish Football Association, said the two clubs had failed to act strongly enough in the aftermath of the game.
He said: “I think people viewed with disbelief the press releases that were issued from both Rangers and Celtic.
“They must be embarrassed, if they could only admit it, as to what went on … they’re the cornerstone of Scottish football, so it really is not in the interests of Rangers and Celtic to be viewed like this.
“That’s why what we need now is an acknowledgement by the clubs that they must put their own house in order.”
He wants to see board members, managers, coaches and players involved in making changes.
“It’s a collective effort and, at the end of the day, this is the best way to solve the problem because my fear is if they don’t, then draconian measures are on the sidelines and they could be deployed quite soon,” he said.
“Taxpayers are paying for policing. They’re also paying to pick up the excesses of violence and of course, for young people, these players are role models.
“Scotland is basically saying enough is enough, but it’s not good enough to just say that it’s up to the Scottish Football Association or the Scottish Premier League, the Scottish Football League or the police or the Church or Alex Salmond -- it’s up to the clubs themselves.”
As well as 34 arrests at the recent game, 187 violent and anti-social crimes were reported, around double the number recorded the Wednesday before.
It has been reported Mr House will deliver a four-pronged plan of attack on the problem at Holyrood tomorrow, including a plan for players and match officials to be given police warnings on conduct before kick-off.
Holding games on Monday nights could also be suggested in a bid to cut down drunken behaviour. A wholesale strategy to eliminate sectarianism from Scotland may be proposed, along with cutting the frequency of Old Firm fixtures.
The Moderator of the Church of Scotland has contributed to the debate, accusing some players, coaching staff and fans at last week’s ill-tempered match of besmirching the game and the name of Scotland.
The Right Reverend John Christie warned the behaviour at the game risks creating another generation for whom “anger and confrontation is the norm”.
“Old Firm games have a long and sometimes tempestuous history, but that should not deter us from beginning to face up to the uncomfortable truth that what has happened in the past, in terms of sectarianism, alcohol abuse, violence -- including domestic violence -- makes what is happening now acceptable,” he said.
Lennon was given 24-hour surveillance at his home and a security officer accompanied him to training on Friday after a hoax bomb addressed to him was found at a Royal Mail sorting office.