The left-back is available for a cut-price fee this summer, with Arsenal able to secure his services for £6 million.
Portuguese newspaper Record states that the Gunners may be intent on securing Grimaldo if they are unable to sign Lisandro Martinez from Ajax.
The latter is also a Manchester United target, and may prefer to play under ex-boss Erik ten Hag.
Grimaldo is 26, but only has a year remaining on his contract, which has reduced his value in the market.
Arsenal manager Mikel Arteta and Grimaldo have the same agency, which could make a deal even easier to conclude.
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