Standard Sport understands that Arsenal have expressed an interest in the Brazilian as Mikel Arteta targets defensive additions following Friday's arrival of former Chelsea winger Willian.
Shkodran Mustafi will miss the start of next season with a thigh injury sustained in the FA Cup semi-final win over Manchester City, while the Gunners could listen to offers for the likes of Rob Holding and Sokratis this summer.
Gabriel is also being monitored by Premier League rivals United, who are looking to find an ideal centre-back partner for captain Harry Maguire.
Manager Ole Gunnar Solskjaer - who typically plays Victor Lindelof alongside Maguire - is understood to be weighing up long-term options from both within - Eric Bailly, Axel Tuanzebe and academy player Teden Mengi - and outside of his current squad, amid links to the likes of Kalidou Koulibaly and with doubts surrounding the futures of Phil Jones, Chris Smalling and Marcos Rojo.
Serie A outfit Napoli and French giants Paris Saint-Germain have also been heavily linked with a move for Gabriel, who almost wrapped up a £30million switch to Everton before the coronavirus pandemic caused football to grind to a halt earlier this year.
The 22-year-old's imminent exit from Lille has now been publicly confirmed by club owner Gerard Lopez, who described Gabriel as one of the two most dominant centre-backs in French football.
A decision on his next club is expected soon.
"The way we work is very simple," Lopez told the BBC World Service's World Football.
"We explain to him and his environment, his agent, what we are looking for, and once we get those offers the choice is his, like we did with Nico [Pepe] and Victor [Oshimen].
"We are there now, so we told him: 'Eventually you have to make a decision but we're not pressing you.' I think he will make it this week, or next week at the latest. He is leaving, we've given the OK for that.
"There's a lot of competition. He's a young player so he's got to make sure he makes the right decision. We help him out a little bit so we tell him: 'We think this might be the right manager or club for you.'
"He's got so much talent, he will succeed wherever he goes."