A banner held up outside Parkhead on Wednesday morning read "Save the ten - time to go Neil", while Celtic fan group, the Green Brigade, also released a statement which called for Lawwell and the board to remove Lennon from his post.
Celtic have claimed just two wins from their last eight matches in all competitions and sit 11 points behind Premiership leaders Rangers in their quest for a record-setting 10th consecutive title, albeit the Hoops have two games in hand.
But Lennon insists there is no panic inside Parkhead boardroom.
"Obviously all of us are not happy with the way things are panning out at the minute," he said.
"But there is a calmness about Peter and the board. There's no panic. We have been over the course and distance many times before.
"I speak to Peter every day. He is very, very supportive, as he is with the players and my backroom team as well.
"So it is as you are, let's get improvement, let's get some wins, let's play the type of football we know we are capable of, which we showed in the last 20 minutes at Easter Road, and we will be absolutely OK."
On the banner, Lennon added: "It is disappointing but fans are entitled to voice their opinion. Whether I agree with it or not is irrelevant, they have put it out there.
"It's something that I am not really going to engage with. I've got a job to do here and I think I've done it pretty well since I came back in."
Lennon, preparing for the Europa League clash with Sparta Prague on Thursday, insists he will not buckle.
He said: "I have sort of reconciled myself with the fact that maybe I am not going to be everyone's cup of tea.
"But it is a great privilege doing this job, the club means a lot to me and it has done for the last 20 years.
"I'm strong, I feel strong, my backroom team are strong and the players will only get better and more consistent as we go along."