The 21-year-old is viewed as a back-up option for Kieran Tierney, who agreed a new five-year contract with the club last week.
Benfica value Tavares at around €12 million but Arsenal are confident of doing a deal for less with the player keen on a move to the Premier League.
Negotiations are continuing for Tavares, who made 24 appearances for Benfica last season including a five-minute substitute outing against Arsenal in the second leg of their Europa League round-of-32 tie in February.
Tavares is viewed as an attack-minded full-back who can make a positive impact in the final third in a similar mould to two other left-backs that the club looked at signing in January: Ryan Bertrand and Patrick van Aanholt.
Bertrand is closing in on a move to Leicester City while Van Aanholt has been linked with Galatasaray. Both players are seeking a chance to compete for regular first-team football but Tavares is younger and willing to wait for his opportunity.
Arsenal need to strengthen cover in the left-back position with Sead Kolasinac deemed surplus to requirements having returned from a loan at Schalke and midfielder Ainsley Maitland-Niles, who has previously deputised as a left wing-back, also likely to leave the club.
Meanwhile, Arsenal have rejected a second offer from Aston Villa for Emile Smith Rowe. The bid was believed to £30m -- a £5m increase on their first attempt -- and dismissed out of hand by Arsenal, who remain in talks to extend the 20-year-old's contract.