The Red Devils invested over £200m during a busy summer transfer window, but have a number of players facing spells on the sidelines due to injury.
United are currently operating without the services of a recognised left-back with Luke Shaw and Tyrell Malacia both on the treatment table.
Erik Ten Hag's side were linked with a move for Chelsea outcast Ben Chilwell on deadline day but opted against a move for the England international, who is not a part of Enzo Maresca's plans this season.
Man United consider January approach for Robinson
According to The Sun, Manchester United see Robinson as a potential long-term successor to Shaw at left-back and have added the Fulham left-back to their shortlist of options.
The 27-year-old currently has a price tag of around £30m and it is understood that Fulham may be open to selling him if they receive an offer that meets their valuation.
The USA international has made a strong start to the season at Craven Cottage, providing two assists in his first three Premier League games, including a superb pass for Adama Traore's equaliser against Ipswich last weekend.
Robinson arrived at Fulham from Wigan Athletic in August 2020 for a relatively low fee of £2m, which means that if the Cottagers receive their asking price, it would represent a significant profit on his sale.
Since joining the club, Robinson has proven to be a dependable defender and has attracted interest from clubs such as Manchester City, AC Milan and Inter Milan.
Fulham's impressive summer transfer window
Fulham had a fairly impressive transfer window, which concluded with Marco Silva's side securing a loan deal for Arsenal attacker Reiss Nelson.
Nelson's move to Craven Cottage marks his third loan stint, having previously gained valuable experience with Hoffenheim in the 2018-19 season and Feyenoord in 2021-22.
Earlier this summer, the Cottagers signed Joachim Andersen from Crystal Palace, Emile Smith Rowe from Arsenal and Sander Berge from Burnely for a combined total of £82m.
Additionally, the club brought in Ryan Sessegnon and Jorge Cuenca on loan from Tottenham Hotspur and Villareal respectively.