However, he is not over the line quite yet, and one other candidate for the position remains very much in the frame.
Former Manchester United and Republic of Ireland star Keane has had the bookies running for cover for the last 24 hours as the money has continued to flood in for him, despite his odds being dramatically slashed.
Keane played for the Hoops for five months in 2006, helping Gordon Strachan's side lift a Scottish Premier League and League Cup double before announcing his retirement from playing at the end of that season.
He moved into management with Sunderland then Ipswich, with varying degrees of success, before accepting the chance to join Martin O'Neill with his national side.
The 42-year-old from Cork is now heavily tipped to make a dramatic return to Celtic as major shareholder, Dermot Desmond, prepares to end the guessing game over who will replace Neil Lennon, who resigned a week ago today.
Keane was due to address a media conference ahead of this weekend's match against Italy at Fulham's Craven Cottage, but O'Neill arrived instead, asking reporters if they were "surprised" to see him.
Asked about Keane's future, the former Celtic manager revealed he has been contacted by the SPFL Premiership champions asking for permission to speak to his assistant.
He stressed that he knew when he took him on with Ireland that offers would come in, but said he expects Keane still to be with the Ireland squad for their weekend friendly.
Celtic issued the following statement on the club website: "Following speculation concerning the manager's position at Celtic, the club can confirm that it is currently continuing its discussions with a number of excellent candidates.
"We will continue this process and select who we regard as the best appointment for Celtic. We will update our supporters as soon as this process is complete."