The 50-year-old Glaswegian has held informal talks with the club over the past couple of days.
However, it is understood Championship club Brighton are also in contact with the former West Brom manager, and any definite job offer from Celtic would have to be forthcoming very quickly.
Clarke has vast experience of coaching at top clubs.
He worked as an assistant manager to Ruud Gullit at Newcastle before taking over as caretaker when the Dutchman left.
Clarke was also assistant to Jose Mourinho, then Avram Grant at Chelsea.
He next moved to West Ham to be assistant to Gianfranco Zola, then joined Liverpool as assistant to Kenny Dalglish.
Clarke finally became a permanent manager in his own right when he accepted an offer from West Brom in June 2012.
He led them to an eighth-place finish in his first season.
But things started to go wrong in his second term, and in December last year he was put on 'gardening leave' until last month.
Clarke's contractual obligations to West Brom have now ended and he is free to take up another job.
He was one of the "five-to-ten exceptional candidates" chief executive Peter Lawwell referred to last week when it was rumoured Keane was about to be unveiled as the new manager of the Hoops.
Keane has spent the last six days considering whether he wants to be considered as a candidate to fill the vacancy created when Neil Lennon resigned a week past on Thursday.
Major shareholder, Dermot Desmond, had made contact with Keane last week, and further discussions took place over the weekend, though sources at Celtic insist he was never actually offered the position.
The bookies were convinced the 42-year-old would be the man, and return to the club where he played for five months at the end of his career in 2006.
But, Keane has now decided it is not for him, and will continue to work on a part-time basis as assistant to manager Martin O'Neill with the Republic of Ireland national side.
While he is preparing to jet out to America with the Irish squad on Wednesday, Celtic have switched their attention to Clarke and to the other names who remain on their list of possibles.
Henrik Larsson had initially been the first choice to be brought back to Parkhead in a move which would have been very popular with a huge section of the support.
But, the former striker is under contract with Swedish club, Falkenberg, until November, and promised them he would not try to get out of that agreement.
However, there was fresh speculation in the last 48 hours that Celtic had made a further approach to Larsson, and the Hoops icon admitted he did not fear the challenge the job presents because he knows what it entails.
He said: "I've seen managers going to Celtic who thought that it was just another club - but it's more than that.
"I think I understand what it means to be the manager there and what is expected of you. And I definitely think I'm capable."
Other names still in the frame include Malky Mackay, Owen Coyle and Brian Laudrup.
But Clarke has now edged ahead in the race to become the next manager at Parkhead, and the next 24 hours could determine whether or not he is the man who follows in the footsteps of triple-championship-winning Lennon.