The 26-year-old is only under contract at the Estadio da Luz for another 12 months, raising the prospect of a cut-price deal in the current transfer window.
Benfica will look to avoid losing Grimaldo on a free transfer in 2023, and Arsenal are supposedly ready to take advantage of the Spaniard's contract situation.
According to Portuguese newspaper Jornal de Noticias - via The Express - Arsenal are now set to ramp up their efforts to sign the defender before the new season kicks off.
The report claims that Arsenal are eyeing a bargain deal of €7m (£5.9m) for the defender, and Benfica would be happy to accept such a figure for the soon-to-be free agent.
Benfica are supposedly expecting Arsenal to make a formal approach for Grimaldo's services within the next week, three years after the Gunners tried and failed to prise him away from the Estadio da Luz.
Grimaldo has also been linked with a return to Barcelona, where he spent eight years in the club's youth academy before signing for Benfica for a £1.9m fee in January 2016.
However, La Blaugrana are now thought to be prioritising a move for Chelsea's Marcos Alonso, which could open the door for Grimaldo to head for North London.
The former Spain Under-21 international has recorded a total of 19 goals and 50 assists in 246 outings for the Portuguese giants, winning three Primeira Liga titles in that time.
Grimaldo ended the 2021-22 season with six goals and nine assists to boast from 48 games in all tournaments, helping Benfica reach the Champions League quarter-finals under Nelson Verissimo.
Benfica are thought to have already made contingency plans for Grimaldo's departure and will move for Borussia Dortmund's Raphael Guerreiro if the Spaniard ends up at the Emirates Stadium.
Arsenal lost their first-choice left-back Kieran Tierney to knee surgery in the spring, which saw Nuno Tavares thrust into the action, but the 22-year-old experienced a hit-and-miss first season in North London.
There have been suggestions that Tavares could be allowed to leave this summer if Mikel Arteta can welcome a new defender to the club, although Brentford have won the race to sign Aaron Hickey.
Ajax's Lisandro Martinez is also thought to prefer a move to Manchester United - where he would reunite with Erik ten Hag - despite being tempted at the prospect of working under Arteta.
Tierney lasted 30 minutes of Arsenal's 5-3 friendly win over Nuremberg in his comeback game on Friday, and Arteta issued a reassuring fitness update on the 25-year-old following his early withdrawal.