The 20-year-old has been a prominent figure in the Guimaraes side this season, starting all 12 of their league games.
He has attracted interest from a number of clubs with the likes of Atalanta and Villarreal in the running.
And the Portuguese club have been informed that the Gunners want to be kept updated on the player should he be allowed to leave.
Bamba is valued at around £26.5million and has been watched by scouts from the London outfit.
The player is not expected to leave in January but there is a feeling that the right offer could change that.
Bamba, who joined Guimaraes in 2020, was born in Italy of Ivorian descent and in May this year was called up to train with a large group of players by Italy boss Roberto Mancini.
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