Gunners boss Mikel Arteta sees the 21-year-old as a perfect replacement for Kieran Tierney.
The Gunners sued him after considering Ryan Bertrand, but the former England international, who left Southampton after seeing his contract expire, appears ready to join Leicester on a free transfer.
Signing a left-back has been a priority for Arteta and sporting director Edu since the January transfer window, when they allowed Sead Kolasinac to join Schalke on loan – while sending Ainsley Maitland-Niles, to West Brom .
Those moves left the club light at the end of the campaign, meaning when Tierney got injured in early April they found themselves without a recognized left-back in some games; Cedric Soares and Granit Xhaka were among those who came in.
Tavares came through the youth ranks of Benfica and played for his B team, before making 41 appearances for the senior team.
He made 14 league appearances last season and five more in the Europa League.
Arsenal are busy in the market this summer as they also pursue a move for Sheffield United goalkeeper Aaron Ramsdale and move closer to Anderlecht midfielder Sambi Lokonga.
It is expected to cost around £ 17million.
Brighton and England defender Ben White is also high on their priority list, as they continue to try to fend off Aston Villa’s advances for Emile Smith Rowe.
The England Under-21 international is wanted by the Villans, but Arsenal are hoping to sign him for a new contract.