Only Jock Stein – 1966/67 and 1968/69 – and Martin O’Neill – 2000/01 – have managed to land the Holy Grail of a domestic clean sweep in Celtic’s trophy-laden 126-year history.
Although Parkhead legend Hay accepts Lennon will always prioritise the SPFL Premiership title and qualification for the lucrative Group Stage of the Champions League at the top of his trophy schedule, he reckons the manager is desperate to enjoy a treble yell.
The double-winning former Celtic gaffer said: “I think one of the big things that will keep Neil Lennon at Parkhead for the foreseeable future is the pursuit of a domestic treble.
“As the Celtic manager you always put winning the league title at the top of your priorities and, of course, it seems almost certain that Neil will be looking to make it four-in-a-row when next season comes around.
"Of almost equal importance is qualifying for the Champions League Group Stage, which is so vital, financially speaking, to the club.
"Again, having done that in each of the last two seasons, Neil will be absolutely determined to repeat the process.
"But Neil is a man with a keen appreciation of Celtic's history and obviously he played a key part in helping Martin O'Neill's side to the treble in 2000/01, so he has had first-hand experience of just how special that achievement is and what it means to the fans.
"With Martin and Jock Stein having been the only two Celtic managers to have achieved that feat I think Neil will be very focused and determined to try and emulate that.
"He has been close with a double behind him last season and I think it is something that will niggle away at him.
"So I just feel that, domestically, that is a piece of unfinished business for Neil that he will very much want to put to bed before his time at the club is completed."
With Lennon already having signed Stefan Johansen and Leigh Griffiths with this summer's Champions League qualification process very much in mind, Hay believes he has already underlined his commitment to the Celtic cause for another campaign.
Hay said: "For me, the signings of Johansen and Griffiths make it plain as day that Neil is already very much down the road in terms of planning for next season. I think he has done very well to get players of their quality in place mid-season.
"You can see there is a real determination to make the Group Stage again and make a better fist of it.
"In terms of down south, would there be an offer on the table for Neil that would include another crack of the Champions League? I'm not so sure.
"Everything points to Neil staying on for a while yet and the work he has already done shows just how hungry he is to make Celtic as successful as possible."