Mikel Arteta has already added goalkeeper Mat Ryan to his squad this month and the Gunners are expected to sign Real Madrid midfielder Martin Odegaard on loan as well.
But the Arsenal boss is looking to add more defensive depth to his options and especially wants to bring in a left-back to compete with Kieran Tierney.
The north London club are monitoring a number of targets and Bertrand is one of the players the club are considering making a move for.
The 31-year-old is out of contract in the summer and he has been protracted talks with Southampton over a new deal.
The defender is happy at Southampton but would be prepared to move on if a new deal is not agreed with the Saints.
However, Southampton are not willing to sell him this month as he is the only senior left-back in their squad at the moment.
The player is happy to wait until the summer and assess his options if he does not sign a new deal with the Saints.