Clement’s team are away from home in the Premiership this weekend, against Dundee United on Sunday (12:00).
Rangers said in a statement: “The board of Rangers Football Club today announces chairman, John Bennett, is stepping down with immediate effect for health reasons.
“The board wants to sincerely thank John for his service and significant commitment over the last nine years and wishes him a full and speedy recovery. It is requested his and his family’s privacy is respected at this time.
“The board thanks John for stepping in at short notice and believes his experience in senior business roles throughout a long and successful career, and also, as a former director of Rangers, will be invaluable at this time.
“The board is undertaking a robust process with the help of professional advisors to fill the roles of CEO and chairperson. In the meantime, the club’s executive management team will continue to lead the club on a day-to-day basis.”