The Hoops are currently 13 points behind Rangers at the top of the Scottish Premiership and have been dumped out of both the Europa League and the League Cup after a sudden dip in form that has seen them pick up just two wins from their last 12 matches.
Even with two games in hand on their Old Firm rivals, the general feeling among many supporters is that the club’s chances of securing a historic 10th consecutive league title are fading rapidly, and there have been widespread calls for Lennon to be relieved of his duties.
Parkhead chiefs have responded to the heightened scrutiny surrounding the Northern Irishman’s tenure by publicly backing him on a number of occasions, but Walker is unconvinced that he is the man to turn things around in Glasgow.
“I don’t think it’s going to work for Neil Lennon”, he told Sky Sports’ Scottish Football Podcast.
"I just see an enormous drop in the level of performance from the players.
"I know how tough it is to make that decision. I was at Celtic in 1991 when Billy McNeil was sacked. You couldn’t think of a better player, better manager, better captain, better Celtic man steeped in their history and tradition. I spoke to him years later, I asked him about that incident and he said, ‘Look, it was time for me to go’.
"He wasn't going to resign, but results were poor. It’s a demanding club, you need to win games. Celtic move on to someone else and they become a great club again – that’s what happens. You take a dip, you make a change, and you look for success again.”
Walker then went on to suggest that the Parkhead board also have to shoulder their share of the blame for the Bhoys’ current situation.
“I’m just not sure the board know what they’re doing”, the pundit said.
"I think they’ve been absolutely gobsmacked by the speed at which all of this has unravelled. Celtic's level of performance has dropped enormously, and at the same time, you’re looking across the road at Rangers – which you always have to do – and Rangers’ level of performance is so high, and it’s so high every week.
"I think there are people at Celtic who have been taken aback by the improvement at Rangers, and that’s where I would criticise them. I could see Rangers were a decent team when Rangers were winning against really good quality opposition in Europe, so it’s no surprise to me that Rangers have taken another step and improved again.
"I think behind the scenes, too many people at Celtic have been laughing at Rangers and just thought it was a stroll towards a tenth title in a row.
"So many things have gone wrong behind the scenes there. I don’t think it’s going to work for Neil Lennon, and eventually, when he does move, I can see some big moves behind the scenes as well.
"I think they’ve been lazy, I think they’ve been incompetent, and I think they have been complacent for this club. You’ve got to remember the financial advantage Celtic have had over Rangers, built up over the last nine titles in a row – it’s enormous. Celtic should be out of sight, and they’re going to play second fiddle.
"Alright, you end up falling out with Brendan Rodgers, you end up not liking him – you get £9 million in compensation for him. In effect, he’s cost nothing.
"To move away from that, to go to someone who is out of work, and very cheap, and who was told under no circumstances to criticise the board – I think that’s when the rot set in, when they made that change.
"Neil Lennon did well to carry on with Brendan Rodgers’ players, they were in a groove, things were coming naturally to them, but the level of performance in the players now has dropped and I can’t see it coming back.”
Protests surrounding Lennon's position with Celtic have turned ugly at times in recent weeks.
As well as clashing with police in the aftermath of the 2-0 defeat to Ross County last month, some supporters have taken to hanging banners outside Parkhead to voice their frustrations.
One particular sign sparked outrage earlier in the week, with the banner reading, "Shoot the board”.
Lennon will again take charge of his struggling side in their Europa League dead rubber against Lille on Thursday evening.
The Hoops host the French side knowing that they cannot mathematically qualify from their group, but will be looking to pick up a positive result to prevent further pressure from being piled on the boss’ fragile tenure.
Celtic drew 2-2 with the Ligue 1 title challengers earlier this season, after squandering a two-goal advantage in the second half.
So far, that result constitutes the Bhoys’ only point on the continent this term – a marked drop-off from the 13 they amassed in the group stage last campaign.
The defending champions’ next domestic outing comes against Kilmarnock in the Scottish Premiership on Sunday.
The Hoops drew 1-1 with Killie earlier in the season, with Ryan Christie’s opener cancelled out by a Chris Burke penalty.