The Spain international has been a long-term target for the Gunners and it was claimed in January that the club have a verbal agreement with the player for a summer transfer.
The speculation has continued to persist and Foot Mercato claim that Madrid are very interested in bringing him to the Bernabeu when the season ends.
Zubimendi has been tipped to join the Gunners for the last few months, but reports claim that he is prepared to prioritise a move to Los Blancos instead.
Madrid are expected to part ways with manager Carlo Ancelotti soon and they have identified Bayer Leverkusen's Xabi Alonso as a possible successor.
Alonso was previously the assistant manager at Real Sociedad and he shared a close relationship with Zubimendi during the same spell.
It appears the duo could be reunited in the Spanish capital this summer.