Tavares's debut season at Arsenal saw him total 28 appearances in all competitions, during which he provided two assists and scored his first goal for the club against Manchester United.
The 22-year-old was expected to undergo a season of development as Kieran Tierney's backup, but he was thrust into the limelight amid the Scot's injury woes.
Tavares's defensive displays were criticised, though, and he was memorably taken off in the first half of the Gunners' FA Cup third-round defeat to Nottingham Forest back in January.
The Portugal Under-21 international also lasted just 45 minutes of April's 3-0 defeat to Crystal Palace at Selhurst Park, and Gunners chiefs were understood to be considering the prospect of a loan move elsewhere for Tavares.
A fee of €40m (£34.4m) would supposedly be required to sign Tavares permanently, with the Portuguese under contract at the Emirates for at least another three years, and there is an option to extend his stay to 2026.
Speculation surrounding a move for Bologna's Aaron Hickey has cooled in recent weeks, and the Gunners are now thought to have identified Martinez as their prime defensive target.
The 24-year-old - who is comfortable in the heart of defence, as a left-back or as a defensive midfielder - is also not due to become a free agent until 2025 and could cost up to €50m (£43m) to prise away from Ajax this summer.
Arsenal must also fend off competition from Erik ten Hag's Manchester United to win the battle for Martinez, who scored one goal and provided four assists in 32 appearances across the Eredivisie and Champions League last term.
Atalanta's head of recruitment Lee Congerton has confirmed that he is an admirer of the former Benfica man, but he is pessimistic about his chances of prising him away from North London.
"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," Congerton told the press.
Arsenal have already welcomed William Saliba back to the club from Marseille to bolster their rearguard, although Les Olympiens are still holding out hope of agreeing a permanent deal for the 21-year-old.