The 60-year-old coach, who worked at Parkhead under Tony Mowbray when Brown was making his start at the club, has been criticised by Brown on the Open Goal podcast.
The former captain said he had “no time” for Grant due to his “zero personality”, suggesting he would have left the club if Grant and Mowbray had stayed.
In response, Grant said on the Go Radio Football Show: “Everyone is entitled to their opinion. I’ve absolutely no problem whatsoever. It doesn’t concern me in the least.
“I’ve had a lot of criticism through my career, so I’ve got to have a big backbone.
“I’ve got a hell of a lot of friends who would say the opposite, but that’s life.
“We move on from it.”
When asked if he could identify the root of Brown’s grudge, he said: “I’m trying to think when it was, it was when I was at Celtic, so 10/15 years ago or something? I’ve not got that memory, obviously he does.
“It doesn’t concern me in the least. If it was Roy Aitken, Danny McGrain, Paul McStay or the great Tommy Burns, it would maybe concern me.
“I’ve never, ever been critical of somebody who’s criticised me. That’s life, I’ve had plenty of criticism.
“One thing is, I will sleep well in my bed tonight. That’s for sure.”
Brown’s remarks about Grant were: “Lenny came in and we started winning games and the momentum changed from when Tony was there.
“I think if Tony and Peter Grant were still there, I’d have been away straight away.
“I had no time for Peter Grant whatsoever.”
When asked about his issues with Grant, Brown said: “Man management. Personal.
“Personality? Zero.
“Tony, I got on alright. But Peter Grant, no.”
