The Gunners want to sign a left-back this summer to provide competition for Kieran Tierney, who signed a new long-term contract last week.
Arsenal have looked at a number of players, including free agent Ryan Bertrand, but Tavares has emerged as a prime target.
The 21-year-old is likely to be available for around £9million, with Benfica ready to cash in this summer.
Arsenal are interested in striking a deal at that price, particularly as Tavares is able to play at right-back also.
The full-back has represented Portugal at youth level and is regarded as a player with a bright future in the game.
Tavares’ arrival would likely accelerate the departure from Arsenal of Sead Kolasinac, who is up for sale this summer after spending the second half of last season on loan at relegated Schalke in the Bundesliga.
