The 18-year-old was a half-time substitute in the Copa Libertadores clash against Huachipato, with a scout from the 20-time English champions said to have been in attendance to monitor his display.
According to the Daily Mail, Man United are keen on Nunes, with the club currently looking at the South American market due to the inflated prices in Europe.
The report, though, claims that Manchester City, Liverpool and Arsenal are also keen on the teenager, with four of the biggest clubs in England therefore potentially battling for his signature.
Nunes has been with Gremio since the age of 15, progressing to first-team level this year, and he has now featured on 13 occasions for the senior side, scoring once and registering two assists.
Man United 'keen on deal for Gremio's Gustavo Nunes'
The youngster, who primarily features as a left-sided attacker, has a contract with Gremio until December 2028, placing the Brazilian outfit in a strong position when it comes to any future negotiations.
Man United currently have two Brazilian players in their squad, but both could potentially be on the move during this summer's transfer window.
Antony is being heavily linked with an exit, having found it difficult to make his mark at Old Trafford since arriving from Ajax in the summer of 2022.
Casemiro also moved to Man United in 2022, and after a strong first season at the club, the 32-year-old has again been a regular this term despite some injury issues, making 25 appearances in all competitions.
However, the ex-Real Madrid midfielder could depart during this summer's transfer window, with a number of clubs from Saudi Arabia said to be interested in his services.
Sir Jim Ratcliffe is expected to oversee a busy summer for Man United
Man United look set to miss out on Champions League football for the 2024-25 campaign, which will undoubtedly impact their finances for the summer market.
However, the club's new minority shareholder Sir Jim Ratcliffe is expected to oversee a busy transfer window, with the Red Devils needing to invest in their squad this summer if they are to enjoy a more successful 2024-25 season.
As many as five players could arrive, with three defenders, one central midfielder and one centre-forward likely to be targeted, while there will also be a number of departures.
Anthony Martial is set to leave on a free transfer, and the club also have decisions to make when it comes to Raphael Varane, Jonny Evans and Tom Heaton, who are all out of contract at the end of June.
Man United have the option to sign Sofyan Amrabat on a permanent basis from Fiorentina this summer, meanwhile, but the Morocco international is expected to head back to his parent club, having found it difficult to impress in Manchester this term.