The Gunners have made multiple bids for the 22-year-old, with the club's improved £25 million ($31m) bid rejected last week.
Though many have expected a deal to eventually get done, Lennon believes that the Scotland international will not be moving to the Emirates this summer.
“The deal, as far as I’m aware, is not happening," Lennon said. "We are delighted about that and it’s just a question of appeasing his mind now.
“As far as I am aware, it is finished. I don’t know if Arsenal are going to meet our valuation. So it is basically as you were."
Lennon has admitted that a transfer falling through could be difficult for Tierney to accept, as the left-back has more than proven himself worthy of a move to a bigger league.
The Hoops boss knows that if Tierney does stick around, he'll have a difficult job to get him refocused as Celtic chase a ninth consecutive Scottish title.
“That may be difficult for Kieran to get his head around – or he comes back to the fold and gets on with things.
“I’ll have to speak to him and I’m sure the club will as well. But it may rumble on as well, I don’t know."
Though Lennon attempted to shut down Tierney's move to Arsenal, he admitted that nothing would be completely settled until after the Premier League transfer window closes on August 8.
“Until the window closes, we can’t relax and I’m sure Kieran can’t either," Lennon said.
Lennon also shut the door on a rumoured incoming move, with former Leicester City full-back Danny Simpson not coming to Parkhead.
“That [Simpson] is not happening," Lennon said. "We are down the line with one player, so he is going through the medical process."