Caretaker Neil Lennon is the man Celtic want to replace Tony Mowbray as their new permanent manager.
And an announcement is expected within the next 10 days.
No deal has been signed with Lennon, but a decision about the 38-year-old former Celtic and Northern Ireland midfielder is understood to be close.
Lennon, who has been in charge since March, has admitted that he may look to bring in an experienced figure to work beside him if he takes on the role.
The former Leicester City midfielder and chief executive Peter Lawwell on Thursday evening began what has been billed as a summer-long series of question and answer sessions with Celtic supporters.
Lennon, who had been reserve team coach, was given temporary control of the team after Mowbray was sacked in March.
But it would be his first managerial job should he be given the role full-time and there have been suggestions that Celtic have been sounding out top names as possible mentors for Lennon.
And he told the meeting of 500 fans convened by the Celtic Trust at Parkhead's Kerrydale Suite: "If I do get the job, maybe I would like an older head to bounce ideas off.
"It is a possibility if the outcome is positive."
Lennon's hopes of being appointed appeared to be dented by Celtic's surprise defeat by First Division side Ross County in the semi-finals of the Scottish Cup.
However, he led the Glasgow outfit to eight straight wins at the end of an otherwise disappointing Scottish Premier League season in which debt-hit Rangers retained the title despite having no funds to buy new players.
Responding to suggetions from fans that Lennon should have been appointed immediately after ending the season with a 100% league record, Lawwell told Thursday's meeting: "We are going through a selection process, which Neil is fully involved in.
"If we gave Neil Lennon the job after the Hearts game because we liked him, that would have been the wrong thing to do.
"But, if he gets the job, it is because he is the best man for it.
"However, it is taking a bit longer than we thought. Bear with us. We should see something in the next week or two."
Lawwell took his "share of responsibility" for the failure of the Mowbray era.
"We brought Tony in for exciting, expansive football," he said. "For whatever reason, it didn't work."
He also admitted that Celtic had lost the SPL title "to a team that is financially crippled" in Rangers.
At the first of the roadshows, which will also take in venues in England and Ireland, leading Celtic Trust figure Eddie Toner received approval from the floor upon calling for Lennon to be appointed immediately.
"The empty seats at Parkhead are alarming," he said, aiming criticism at the chief executive, chairman John Reid and major shareholder Dermot Desmond.
"I think you will have a very hard job selling season tickets.
"I can't put my faith in John Reid, Dermot Desmond or you.
"But, if the club was to come out and appoint Neil Lennon tomorrow, he will be cheered down London Road."
