Lennon is believed to rate Tierney at £25 million ($31 million) but Arsenal's latest bid is reported to have fallen short of that price tag.
Celtic plan to stand firm in their valuation of the Scotland international and Lennon on Tuesday said: "I don't know what Arsenal's financial state is, I know what ours is and we are quite comfortable with the situation at the moment.
"There has been a second bid, the bid has been rejected, and it's as you were. Whether they come in again or not, I can't answer that.
"It doesn't surprise me there are potential suitors out there for a player of his quality but he's our player, he's on a long-term contract and we don't want him to go."
Lennon was asked if Arsenal's bid was disrespectful given Manchester United recently paid Crystal Palace £50 million for young right-back Aaron Wan-Bissaka.
"Disrespectful is strong but I do feel our valuation of Kieran is correct," Lennon said. "We can't do anything about what clubs do in England. Fifty million for Wan-Bissaka is a lot of money.
"We feel Kieran is a far more experienced and more rounded full-back at this juncture of his career. So he's an asset for us and we rate the player very, very highly.