Lennon is planning talks with the 22-year-old left back after receiving feedback that the English Premier League club will not return with a fresh offer which matches Celtic’s £25 million price tag for the Scotland international.
Arsenal have already had two bids rejected by the Scottish champions, who were unhappy at the nature of the clauses involved in fulfilling their valuation of the player.
Tierney had been preparing himself for the likelihood of leaving his boyhood club but, while Lennon has not ruled out the saga “rumbling on” for the rest of the transfer window, he wants to ensure Tierney is able to refocus on remaining at Celtic.
“As far as I am aware, it is finished with Arsenal,” said Lennon. “That’s the vibe I’m getting. I don’t know if Arsenal are going to meet our valuation. I would have thought they would have come back in by now. So it is basically as you were.
“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 about it and I’m sure the club will as well. But it may rumble on as well, I don’t know. Arsenal might be the first of many who raise their head above the parapet.
“Until the window closes, we can’t relax and I’m sure Kieran can’t either.
“I know, having been in that position myself, how unsettling it is. There’s no question about that,” added Lennon.
“Sometimes your mind races away with you. You try to tell him to just stay in the present and let things take their natural course but it is difficult when you are young and ambitious.
“My advice to him is just let it happen if it is going to happen. If it doesn’t, then try to re-engage yourself with what you are good at, which is being around Celtic.”
Tierney is still sidelined as he recovers from the double hernia operation he underwent in May.
Lennon continued: “He is still doing his rehab, so it has been difficult for him from that aspect as well.
“He is frustrated by the injury and the pain he is still getting from the pelvic area.
“He is grounded, he has done a lot in his career already. But you still have to remember he is only 22 so he hasn’t really grown into his clothes yet, if you know what I mean.
“Until you get to 25, then there is still a boyishness about you at times.
“Getting fit again is the first step he needs to take before anything else happens. He will be happier in his own mind once he does that. He is trying his best to do that at the minute.
“It is difficult anyway. He is not one of those fly-by-night players who comes in and is mercenary. Kieran has been in the system and you know what the club means to him.
“It would be a huge wrench to him and his family if he were to go and he has made that pretty clear.
“Again, we are talking hypothetically here. The deal, as far as I’m aware, is not happening. We are delighted about that, obviously, and it’s just a question of appeasing his mind now.”
Lennon confirmed that Celtic are close to completing a £1.6 million deal for Israeli international Hatem Abd Elhamed from Hapoel Be’er Sheva after securing a work permit for the 28-year-old.
Elhamed can play in central defence or right-back, where he could be a replacement for Mikael Lustig following the Swede’s summer departure. But Lennon, who confirmed a possible trial period for former Leicester City right-back Danny Simpson has been cancelled, says Celtic are also closing in on another target in that position.
Lennon, meanwhile, expressed his desire to hold onto Jozo Simunovic amid interest in the Croatian central defender from French club Lille.
“There appears to be interest in Jozo but nothing concrete so far,” added Lennon. “He’s a player I don’t want to lose. He’s been fantastic for me since I came back to the club. He has a year left on his contract and the club have an option to extend it, so we are still in a very strong position on that.”