Former midfielder Keane has emerged as the leading candidate to replace Neil Lennon in the dugout, although chief executive Peter Lawwell confirmed yesterday that the club still have a shortlist of up to 10 names under consideration.
The likes of Henrik Larsson, Malky Mackay and Owen Coyle have also been touted for the position but it is Republic of Ireland assistant manager Keane who is inching towards the Parkhead hotseat.
Hay tipped Keane to succeed Lennon in SportTimes last week and the former boss knows the issue can't be left unresolved for too long.
He said: "There is a balance between getting the right man and making the appointment at the right time.
"Just now, there is no real rush. The players are not back for a few weeks yet. Celtic won't rush into it, they can't be too quick and not get the man they really want.
"But, having said that, they will want someone in fairly soon and certainly before the start of pre-season.
"I would imagine it would be a maximum of another two weeks before we get an appointment. It might even happen next week.
"I wouldn't think the board would like to keep it going much longer than that.
"Sooner rather than later all the speculation will have finished and we will know who the new manager is."
It is now over a week since Lennon shocked the Parkhead faithful and announced that he would be stepping down from his position after guiding Celtic to their third-straight top-flight crown.
A series of high-profie contenders have been touted for the Hoops top job so far but it is Keane who looks set to given the green light by the Celtic hierarchy in the coming days.
Lawwell said: "We are looking at a number of exceptional candidates and we're going through a process. Hopefully it won't take too long.
"Clearly we want to do it as soon as we possibly can but these things you can't put a time on.
"We have got a number of excellent candidates, Roy is one of them and we'll see what happens in the next week or two.
"He's played here, he's a Celtic supporter, he's a first class guy and like the others are, as I said we have between five and 10 candidates.
"You just have to do your work properly and hopefully we'll get it done as soon as we can.
"As soon as we do that we will let our supporters know.
"You don't want to get it wrong by going too fast, you need to take your time and make sure you get the right guy."