Celtic let Tierney leave on the final day of the English transfer window for £25million.
The left-back joined Arsenal after a summer of uncertainty, having been at Celtic since the age of nine.
The Scottish champions rejected multiple bids for Tierney over the summer, before finally letting him go once their asking price was met.
Celtic had initially been chasing Brentford left-back Rico Henry as a replacement, but the 22-year-old signed a new contract at the Championship side.
Henry had been tipped to head north for a fee of £7m, but instead opted to stay at Brentford.
Instead, Lennon has identified Kilmarnock defender Greg Taylor and Rosenborg’s Birger Meling as two potential signings.
Celtic are keen to bring in a new defender before Monday’s deadline, but were apparently unwilling to pay Kilmarnock’s asking price for Taylor.
Killie demanded £3m for Taylor, which Celtic bosses decided was too steep.
Instead, they are willing to pay £1m, along with Jack Hendry on loan, though it seems unlikely that they will be able to complete a deal.
Lennon is also keen to bring in a right-back, and is reportedly targeting Olympicaos right-back Omar Elabdellaoui.
However, Celtic’s Champions League qualifying defeat made the move less likely.
The Bhoys were beaten by Cluj over two legs and failed to qualify for the tournament.
Olympiacos, however, have one foot in the Champions League, after defeating FC Krasnodar in the first leg of their qualifying match.
For that reason, it is possible that Elabdellaoui will snub a move to Parkhead in order to continue playing in Europe.
Lennon is also interested in bringing in a right-back, a midfielder, and a striker.