Zubimendi is continuing to impress in LaLiga, and has been earmarked as a long-term replacement for Barcelona midfielder Sergio Busquets.
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The Catalan giants have been long-term admirers of the 24-year-old, as Xavi looks to bolster his midfielder this summer.
According to journalist Eugenio Mateo Serrano, though, Arsenal could pip them to his signature.
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"Xavi needs another player instead of Busquets. The number one pick is Martin Zubimendi, from Real Sociedad," Serrano told Calciomercato.
"But it's also difficult because Zubimendi wants to play in the Premier League, and for me Arsenal are in pole position."
It looks as though the Gunners have stolen a march on Barca, as Mikel Arteta seeks long-term cover for Granit Xhaka and Thomas Partey.
Arsenal brought in Jorginho in January, but more midfield signings are likely in the future.
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It is believed the north London club would have to pay the €60 million [£53m] release clause in Zubimendi's contract if they are to get their man.
According to SPORT, Barcelona have worked tirelessly on trying to lure the player to the Camp Nou in the past, and Sporting director Jordi Cruyff is believed to have a good relationship with the player's agent.
Interestingly, though, Zubimendi does share the same agent as Arteta, and if the player decides the Premier League is his preferred destination, Arsenal could get their man.
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It seems as though Barca do need to act fast if they are to sign the player, otherwise they risk losing out to the Gunners.