Robinson made the move to Fulham from Wigan Athletic in 2020, and he has been a star performer for the London club, making 162 appearances in all competitions, scoring three goals and registering 15 assists in the process.
The 27-year-old has three assists in 11 Premier League appearances this season, and there has been a host of speculation surrounding a possible move away from Craven Cottage next year.
Fulham are in a strong position when it comes to the defenders future, though, as he signed a new contract in July 2023, with his terms due to run until the end of the 2027-28 campaign.
Brown has said that Man United and Liverpool are both interested in signing the United States international next year, but Fulham will refuse to let him leave during the January transfer window.
Robinson 'on Man United and Liverpool's radar'
"I think Robinson is a really good player," Brown told Football Insider. "If ever there was an example of a player who has improved, it's him. There were big questions at one stage about his defensive abilities.
"But he's improved in that department and he's so dangerous going forward. After his form so far this season, I've heard there are a few clubs keeping tabs on him.
"Some of the top clubs in the Premier League have been watching him for a while, but the way he's playing at the moment, that interest has grown. He's one of the leading candidates for any team looking to sign a left-back.
"I'd count Man United and Liverpool among them. Whether they'll make a move or not, I'm not sure, but they're among the clubs tracking him. From what I hear, Fulham won't let him leave in January anyway, so if it happens it won't be until the end of the season."
Have Man United's left-back plans changed?
Man United will not make any decisions at this stage of proceedings, but it is certainly possible that their left-back plans will change if both Luke Shaw and Tyrell Malacia are able to prove their fitness in the coming months.
Neither has made a first-team appearance this season, with Shaw battling a calf problem, while Malacia has not featured for the senior side since the end of the 2022-23 campaign due to a severe knee problem.
Shaw is now back in full training, though, while Malacia turned out for the Under-21s against Huddersfield Town in the EFL Trophy earlier this week, with Man United being handed a major double fitness boost.
There are concerns when it comes to the long-term fitness of the duo, though, and Man United are still believed to be keen to sign a new left-back next year, with Bournemouth's Milos Kerkez also allegedly admired by Ruben Amorim's side.