A number of top clubs including Manchester City, Napoli and Inter Milan have been linked with the Spanish left-back.
And according to Mundo Deportivo, Arsenal and Atletico Madrid are frontrunners.
The Gunners are said to be weighing up an offer for Grimaldo, 23, which could involve Nacho Monreal moving in the opposite direction.
While Atletico are in the market for a new left-back with Filipe Luis out of contract in the summer and Lucas Hernandez linked with an exit.
Arsenal boss Unai Emery wants to bring in at least FIVE players but will have just £40million to spend this summer if the club miss out on a Champions League spot.
The Spaniard wants two wingers, a couple of full-backs and at least one centre-back.
Even with a top-four finish, Emery's budget would only be about £60million.
Benfica want about £26m for Grimaldo and it is that price-tag which stopped him joining Napoli last summer.
Atletico believe they can land him for a cut-price fee, while Arsenal would also look to negotiate a cheaper fee.
Grimaldo came through Barcelona's famed La Masia academy and left Barca B for Benfica in 2016.
He has scored five goals and set up four more in the Portuguese league and Champions League this season.
And the defender, who represented Spain at youth level, looks to have a bright future.
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