The Scottish giants are on the lookout for a new boss after sacking Neil Lennon in February.
A number of candidates, including Keane, are believed to have held talks over succeeding Lennon.
According to The Sun, however, Celtic have yet to get back in touch with the Irishman.
Former Celtic midfielder Keane is said to have accepted that he will not be appointed, with Eddie Howe the bookies' favourite.
The same report indicates that the 49-year-old, who has previously managed Sunderland and Ipswich Town, split the opinion of Celtic's hierarchy.
While some liked the passion showed by Keane, others supposedly felt that a more modern manager is needed.