The 22-year-old spent 11 years at Real Madrid and made just 10 senior appearances before joining Girona in the summer of 2022, with Los Blancos retaining 50% of his rights as well as a sell-on clause.
After making 34 La Liga appearances in his first season at Girona in 2022-23, Gutierrez has since started 15 top-flight games this term for Michel's side, who surprisingly sit two points clear at the top of the table.
The Madrid-born defender scored his first goal of the campaign in Girona's impressive 4-2 triumph against reigning La Liga champions Barcelona at Camp Nou last weekend.
Gutierrez is under contract at Estadi Montilivi until June 2027 and is understood to possess a €35m (£30m) release clause, with Real Madrid entitled to half that sum if another club activates the clause due to their aforementioned sell-on agreement.
Although Los Blancos are said to be closely monitoring Gutierrez's progress, Sport claims that they are not considering a move for the left-back in the near future as they have eyes on other transfer targets.
It is thought that Carlo Ancelotti's side are instead prioritising a summer transfer for Bayern Munich left-back Alphonso Davies and are keeping a close eye on the Canadian's contract situation.
The report adds that Man City now present a threat to Real Madrid in the race for Gutierrez, with the Citizens said to be keen to secure the services of the highly-rated defender.
Man City may have an advantage in their pursuit of Gutierrez as they are one of 12 teams along with sister club Girona who are a member of City Football Group, who owns 47% of the Spanish side.
Interestingly, Girona's chairman Pere Guardiola is the brother of Man City manager Pep Guardiola, but whether that relationship can favour negotiations between the two clubs regarding transfers remains to be seen.
After allowing Joao Cancelo to join Barcelona on loan in the summer, Man City spent big to recruit Josko Gvardiol from RB Leipzig, and the Croatian has competed with Nathan Ake and Sergio Gomez for starts at left-back in Guardiola's side this season.
Man City, meanwhile, have also been tipped to sign a new midfielder in January if they agree to part ways with Kalvin Phillips, who is widely expected to depart having failed to gain regular first-team football at the Etihad Stadium.