The 22-year-old arrived last summer to provide backup to Kieran Tierney but was thrust into the first-team amid the Scotland international's frequent injury problems.
Tavares ended his debut season at Arsenal with one goal and two assists from 27 appearances in all competitions, but his performances earned mixed reviews.
The left-back showed flashes of quality going forward, but some shaky defensive displays were criticised, and he was taken off early by Mikel Arteta in defeats to Nottingham Forest and Crystal Palace.
The Gunners are said to be considering the possibility of sending Tavares out on loan, and Atalanta chief Congerton has confirmed reports claiming that his side hold an interest in the Portugal Under-21 international.
However, with the defender under contract at Arsenal until the summer of 2025 with the option of an additional 12 months, the Atalanta chief is not expecting an easy period of negotiations.
"I've followed Tavares since he was playing for Benfica's youth team, he's a strong player. But Tavares is under contract with Arsenal - and it's really difficult to sign players from Premier League clubs," the Evening Standard quotes Congerton as saying.
Tavares started 13 Premier League games for Arsenal in the 2021-22 season, and the North London club are said to be after a €40m (£34.5m) fee to sell him permanently.
Arsenal's defence has already been bolstered by the return of William Saliba from Marseille - who have also been linked with Tavares - and the Gunners are being tipped to push ahead with a deal for Ajax's Lisandro Martinez.
The Argentine has experience in the left-back area - which could therefore push Tavares further down the pecking order - but the Gunners will likely have to battle Manchester United for his signature.
Arteta and sporting director Edu have focused on bringing in youthful players in the early stages of the summer transfer window and have concluded deals for 19-year-old Marquinhos and 22-year-old Fabio Vieira.
Vieira's arrival from Porto in a €40m (£34.5m) deal was confirmed on Wednesday afternoon, and the midfielder has turned out alongside Tavares for Portugal's Under-21 side.
Arsenal should also announce the arrival of goalkeeper Matt Turner from the New England Revolution in due course, and negotiations are thought to be progressing well with Manchester City over a move for Gabriel Jesus.