The 22-year-old has been pivotal to Girona's remarkable title push in the 2023-24 La Liga season, contributing one goal and six assists in 33 matches across all competitions.
Gutierrez - who is comfortable as an out-and-out left-back or higher up the field - has started 28 La Liga games for Michel's side, who sit third in the table with 65 points from 30 matches.
The Spaniard - a one-cap international for La Roja's Under-21 team - spent 11 years on the books at Real Madrid from 2011 to 2022, but he played just 10 games for Los Blancos before being shipped out to Girona.
The La Liga upstarts have Gutierrez under contract for another three years, but both Man United and Arsenal will supposedly test Girona's resolve with bids in the summer transfer market.
Sir Jim Ratcliffe 'driving interest in Gutierrez'
According to The Mirror - via TEAMtalk - Man United are bidding to steal a march on their London foes through Sir Jim Ratcliffe.
The INEOS founder has supposedly identified Gutierrez as a top candidate to solve the Red Devils' left-back problems, and he is prepared to fork out the big bucks to win the race for the Girona man.
Luke Shaw is Man United's first-choice on that side when fully fit, but the England international has frequently struggled with injuries and is currently sidelined with a severe muscular injury.
Meanwhile, Tyrell Malacia has not made a single appearance this season due to a knee operation, forcing Diogo Dalot and Aaron Wan-Bissaka to fill in on the left-hand side throughout the campaign.
With Wan-Bissaka's future also shrouded in uncertainty, a new left-back is expected to be a top priority for Ratcliffe in the summer market, although there could be another hurdle for them to overcome in their chase for Gutierrez.
Man United's Gutierrez chances hinge on Alphonso Davies future?
While the 22-year-old failed to make the grade at the Bernabeu, Real and Girona supposedly shook hands on a buyback clause worth just €8m (£6.9m) when Gutierrez made the move to the latter two years ago.
As a result, Los Blancos have first refusal on Gutierrez's future, and Carlo Ancelotti's side are known to be in the market for a new left-back this summer amid Ferland Mendy's struggles for fitness.
Rather than give Gutierrez a second chance, though, the La Liga leaders are pursuing Bayern Munich's Alphonso Davies, who has hit a contract impasse with the German champions and can leave for free in 2025 as things stand.
Even if Real Madrid are successful in their attempt to sign Davies, the report adds that they may still opt to trigger their cheap buyback option for Gutierrez before demanding a heftier fee from Man United.
Should Los Blancos fail to prise the Canada international away from the Allianz Arena, though, they could instead decide to re-sign Gutierrez and keep the 22-year-old on the books for good.