After winning nine successive league titles in a row, Celtic lost their grip on the Scottish Premiership this season as runaway leaders Rangers were crowned champions last month with six games to spare.
Celtic have been without a permanent manager since Neil Lennon left the Hoops in February after falling 18 points below Rangers at the summit of the Scottish Premiership.
Former Ipswich manager Keane was touted as a potential candidate for the top job at Parkhead but the Irishman is set to miss out on the position as Celtic are in advanced talks with ex-Bournemouth boss Eddie Howe.
It is claimed that Celtic wanted a manager with both a recognisable name and a track record, which meant that Keane taking the reins was not a likely prospect.
The Irishman’s last stint as manager ended in 2011 when he resigned as Ipswich boss before he took up assistant manager roles at the Republic of Ireland national team and Aston Villa.
Howe is the top candidate to land the Celtic job after proving his managerial prowess with Bournemouth in the Premier League.
Howe, whose playing career was cut short by injuries, has a proven track record of developing young players and selling them on a profit, which has also been a strategy for Celtic in recent seasons.