The 22-year-old came through the youth system at Real Madrid, turning out on 10 occasions for their first team, before making the move to Girona in August 2022.
Gutierrez was in strong form for the Catalan side last season, making 36 appearances in all competitions, while he has also impressed this term for La Liga's surprise package.
The defender has contributed one goal and three assists in 20 outings this term for a team that are currently second in the La Liga table, only behind Real Madrid on the head-to-head record.
Gutierrez again played the full 90 minutes of Girona's 4-3 win over Atletico Madrid on Wednesday, and according to The Sun, Man United had scouts at the match to watch him in action.
The report claims that the 20-time English champions are planning to sign a new left-back this month, having allowed Sergio Reguilon to return to parent club Tottenham Hotspur.
Girona would reportedly listen to offers of around £22m for Gutierrez, but the move would be complicated by the fact that Real Madrid retain 50% of his rights and also have a sell-on clause.
There have been suggestions that Los Blancos are also keeping an eye on his performances amid the speculation surrounding Ferland Mendy's long-term future at Bernabeu.
Gutierrez's form is yet to earn him a senior call-up for Spain, but he has represented his national side at Under-17s, Under-18s, Under-19s and Under-21s level.
Man United have had problems at left-back this season, with Tyrell Malacia yet to be involved during the 2023-24 campaign due to a knee injury which required an operation.
Luke Shaw is the first-choice left-back at Old Trafford, but he has only made 10 appearances this season due to injury problems.
The 28-year-old returned from a three-month absence due to a muscular problem against Everton at the end of November, but he has missed his side's last two games with an unspecified issue.
Shaw could again miss out against Wigan Athletic in the FA Cup on Monday night, which is likely to lead to Diogo Dalot again being pressed into action at left-back.
Man United also have two left-backs out on loan, with Alvaro Fernandez with Granada in La Liga, while Brandon Williams is currently representing Ipswich Town in the Championship.