The Daily Mail says Keane is one of a list of candidates the Parkhead club have already spoken to.
In a statement earlier this week largest shareholder Dermot Desmond vowed to take time to appoint the right man, insisting the process of 'research, investigation, interviewing and negotiation' is underway.
A Celtic player during the 2005-06 season, Keane has held discussions with senior figures leading the search.
His last frontline role in management was at Ipswich Town a decade ago - the 49-year-old has since worked as assistant manager to Martin O'Neill with the Republic of Ireland, Aston Villa and Nottingham Forest.
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