George Mudie, Labour MP for Leeds East and a former leader of Leeds City Council, told the Yorkshire Evening Post: "Given the fiasco that is Leeds United at the moment it is inappropriate for him to be involved with our education services."
Coun Andrew Carter, Tory leader, said: "He now has a significant image problem. I do not think that's in the best interests of Education Leeds or the city's schools. It might be wise for him to consider his position."
Insiders at Education Leeds however insist he still has the full confidence of its senior staff. A council spokesman today declined to comment on the criticism of Mr Ridsdale, saying only: "He continues to fulfil his role as chairman."
Mr Ridsdale was unavailable for comment.