The Irish legend has shot to the top of various bookmakers’ list, ahead of Chris Wilder, John Kennedy and Enzo Maresca.
It was reported earlier this month that Keane, who played for Celtic during the 2005/06 season, was eyeing a return to management with the vacant Parkhead post.
Kennedy has been working in an interim role since Neil Lennon’s departure back in February.
Keane, as low as 1/3 with Skybet, last managed a team in 2011, an 81 game spell with Ipswich Town. The seven-time English Premier League winner with Manchester United as a player has also had coaching spells with Ireland, Aston Villa and Nottingham Forest, working alongside former Celtic boss Martin O’Neill.
The 49-year-old currently works as a pundit for Sky Sports.
Earlier this week, Celtic’s principal shareholder Dermot Desmond provided an update on the manager search, however he kept his cards close to his chest.
He said: “We are mindful of the fact that there is a huge amount of interest and speculation among our supporters and the press.
"However, we have to respect the privacy of potential candidates and the integrity of the process, and so I think it would be inappropriate to make comment on the type of candidate we are looking for.”
Meanwhile, former Liverpool and Newcastle United manager Rafa Benitez all but ruled himself out of the running for the Celtic job, revealing his “priority" is a return to England.