The former Liverpool boss has been heavily linked with the Foxes job after Claude Puel was sacked on Sunday.
But Hartson, who spent five years at Celtic Park between 2001 and 2006, believes the Northern Irishman will resist any advances to focus on leaving a legacy with the Scottish champions.
“I can see why they would want him and why he is still very attractive to clubs in England but I am stubborn and I am going to hold true to what I genuinely believe here – I don’t think Brendan Rodgers is going anywhere,” Hartson told the Evening Times.
“I’ll stress that is an opinion and nothing more. I have not asked him. I have not spoken to him or anyone close to him.
“But what I will say is that in any dealings I have had with Brendan in recent years he has always stressed just how happy he is in his work at Glasgow. He loves the club.
“He is settled in a nice area just outside the city and he has always spoken of just how much he has enjoyed his life in Glasgow.
“I see the attraction of Leicester. They are very wealthy. They play in one of the top leagues in the world and there would be money to spend.
“But I would still argue that in terms of clubs, Leicester are nowhere near Celtic. And I say that with the greatest respect.
“And I know I have said it before, but for me it is vital for Brendan to leave a legacy at Celtic.
“He can be the man not just to take them to 10-in-a-row but to 11, to 12, to 13, to 14… if the board go on and back him this summer then it could be anything.
“There could be no let up to just how dominant Celtic are in a domestic sense.”