Gunners boss Mikel Arteta revealed this week the club are in the market for a left-back to compete with Tierney after Sead Kolasinac was allowed to join Schalke on loan.
Arsenal are looking at a number of targets, but it is understood that Bertrand is one player being monitored.
Bertrand is into the final six months of his contract with the Saints and has been in protracted talks over a new deal with the south coast club.
The former Chelsea full-back, who still lives near the Blues’ west London training ground and commutes to Southampton, was of interest to Leicester in the summer following the sale of Ben Chillwell and the Foxes have kept tabs on the situation since.
The 31-year-old is thought to be happy to stay at the club he joined in 2014 - but, should a new deal not be agreed, offers from the likes of Arsenal would be hard to turn down.
Southampton would not sanction a sale this month with Bertrand being their only senior left-back, while the player would be happy to wait until the end of the season to assess his options if the new contract is not signed.
Southampton manager Ralph Hasenhuttl has often said that he only wants players who are entirely committed to the club to sign new deals and they are also in talks with striker Danny Ings and goalkeeper Alex McCarthy over a new contracts.
Tottenham have held an interest in Ings, but asked about any move today, manager Jose Mourinho said: “I refuse totally to say any word about him because he is a Southampton player and I respect that.”
Arsenal are scouting and looking at a number of left-backs and are one of several Premier League sides watching Crystal Palace’s Tyrick Mitchell.
The 21-year-old has broken into the Eagles’ first team and he has impressed after being handed the chance to shine.
Arsenal, along with other Premier League sides, have scouted Mitchell and are monitoring his situation at Palace.