The 50-year-old Ayrshireman has jumped to the top of the Hoops' list to replace Neil Lennon. Clarke was the subject of sustained support in the betting as Roy Keane informed Dermot Desmond he no longer wished to be considered for the vacancy.
Keane was later reported to be on his way to Aston Villa as Paul Lambert's assistant while another in the running, Oscar Garcia, accepted a job in Israel with Maccabi Tel Aviv.
Clarke played for St Mirren and Chelsea before working as assistant manage to Jose Mourinho, Avram Grant, Gianfranco Zola and Kenny Dalglish, and finally became a manager in his own right in 2012 when he took over at West Brom.
He was put on gardening leave in December, but is finally free to take up another position.
Brighton are set to speak to him about the manager's job there, but Celtic have a head start.
Sources at Parkhead insist they still have some way to go in their negotiations - Malky Mackay also remains a strong candidate - but the idea of Clarke becoming the next boss is already winning support among the fans.
However, it is not the first time moves have been made to recruit him.
As former boss, Davie Hay, explained, Celtic wanted to sign Clarke when he was starting his career as a full-back with St Mirren, only to lose out to Chelsea, where he went on to play for 12 years.
Hay recalled: "Danny McGrain was coming to the end of his career, and we identified Steve as a young player who could take over from him at right-back.
"He certainly looked to have a bright future, but our offer was not accepted because Chelsea came in with more money.
"Alex Smith was the Saints manager at the time, and, coincidentally, I met him last week and he was reminding me of how we missed out on Steve."
Hay, who had been a supporter of the Hoops' move to make Keane their next manager, believes Clarke would bring a lot of the qualities he had as a player to the Parkhead hot seat.
"Steve has always been very professional in everything he has done, as a player and since he moved into management," said the man who played for Celtic and Chelsea and was dubbed the Quiet Assassin.
Whoever becomes Celtic manager will find games are already arranged.
They will play German Third Division side, Dynamo Dresden, in the Glucksgas Stadium on Saturday July 19, and St Pauli, in Hamburg, a week later.
Meanwhile, former Celtic striker Gary Hooper has rocked Norwich by telling the club he wants to quit Carrow Road this summer.
The Englishman scored eight goals for the Canaries following his £5million switch from Parkhead twelve months ago but couldn’t prevent City being relegated to the Championship.
Hooper is determined to continue his career in the Premier League and has told the Norwich hierarchy he is looking to leave during the close season, with newly-promoted Queens Park Rangers and Paul Lambert’s Aston Villa already showing an interest in the want-away frontman.