The Scottish Premiership champions were left stunned back in February when Brendan Rodgers walked away in controversial fashion with three months of the season remaining, securing a return to England with Leicester City.
Since then, Lennon has been prowling the touchline, taking charge of his former side on an interim basis until the end of the season.
The 47-year-old has done a fine job since then, leading Celtic to the semi-finals of the Scottish Cup while opening up a 10 point lead over Rangers in the league.
And Hartson believes the right man for the job is already in the Parkhead dugout.
“I have heard one or two people voice a belief that they would prefer the club to go elsewhere but I think that is unfair. For me it would be Lennon’s job every day of the week and I will be interested to see just how that plays out,” Hartson, who played alongside Lennon in the early 2000s, said.
“He has been ridiculously successful as a manager at Celtic, especially in Europe. Let’s not forget that it was Lennon who got a win over Barcelona at Celtic Park.
“He knows this club inside and out and he is a very astute and intelligent guy.”
Celtic have been linked with a number of alternative options with The Mail reporting that Alex Neil, the former Hamilton boss who has done a fine job at Preston North End, is on their radar.