Roy Keane, the man I tipped for the job last week, now appears to be the front-runner and I would expect an announcement on the new boss to made fairly soon.
It is then that the hard work will begin as Roy, if he does indeed get the nod, gears up for his first season in the dugout at Celtic Park.
The positive thing for whoever is given the green light by the Celtic board is that the squad the new man inherits does not need radical surgery to be successful in Scotland.
But I am sure money will be spent to improve the group for another crack at the Champions League.
The manager will have his own thoughts on the squad, players he would like to bring in and the ones he will try to move on during the summer. There is not a major overhaul needed, that is for sure.
The big thing for the new manager - and Celtic - will be holding on to the likes of Fraser Forster and Virgil van Dijk, both of whom were excellent this season and who will attract interest from other clubs. That was going to be a problem even if Neil had stayed on.
If Fraser gets a chance at the World Cup and does well, it could make it even harder to keep him next season.
You always want to hold on to your best players and the new boss will want to bring in his own players as well.
The manager will have a couple of weeks to assess the squad before Celtic play their first Champions League qualifier in July and that will be a really important time for him as he looks at what he has got and what he needs.
It is not as if he is coming in mid-season and won't be able to use the transfer market. Whoever gets the job will know the squad and the players and will have his ideas even before he gets the job.
He will be aware of the make-up of the squad and the quality of the group of players Neil has left him. And he will be able to speak to the coaches and people at the club about the best way forward.
He can't afford to be rash in the transfer market and he will make the most of the opportunity to see them training and playing in friendly games before deciding what to do in terms of arrivals and departures.
The new boss may well have different thoughts to mine but I would say the forward area will be one that will see some investment this summer.
Celtic have good players up front, but they could do with someone slightly different in style who can compliment Kris Commons, Anthony Stokes and Leigh Griffiths.
The squad is strong in midfield, but there could be room for another player in there and they could look for cover at centre-half and left-back, although there are players that can drop in to those positions and do a good job.
T HE main areas for me are a striker, then a creative central midfield player, and then a central defender and left-back.
It will be a busy few weeks for the new manager and his staff and an exciting time for the players as well.
Players know that these things happen in football, that team-mates and managers come and go and you just have to get on with it and do your job.
There are players that have been at the club for a while and everyone in the squad will be keeping a close eye on the developments surrounding the new manager.
It will help him that the squad have been together and been successful for a few years and he is going into a good dressing room.
There are players that didn't feature as much as they would have liked under Neil last season and they could see the change in the dugout as a chance for them to establish themselves at Celtic.
For whatever reason, there are players for whom being at the club has not worked.
There is a chance they will be given an opportunity and could catch the manager's eye but, from what I have seen of the fringe players it is they, not the manager, who have to change.
Some may be looking at this as a fresh chance for them to impress and make a career for themselves at Celtic, but a lot of them didn't impress enough last season.
And there wasn't enough on show to convince me that, a new manager or not, they can become first-team regulars next season.
It is shaping up to be a crucial and exciting summer for Celtic and their fans, and I would expect the ball to start rolling sooner rather than later.