The INEOS chief will reportedly review United’s football operations before making any required changes.
Chief executive Richard Arnold and football director John Murtough are among those facing uncertain futures, but Ten Hag is said to retain the faith of United figures for now.
The Red Devils have lost six of their opening 12 games this season, winning just once by more than a one-goal margin.
It translated to the club’s worst league start since 1989 before recording back-to-back wins against Brentford and Sheffield United.
And Potter could emerge as a surprise candidate to replace Ten Hag if United’s problematic campaign takes another turn for the worse in the coming months.
Click here to join our Man Utd WhatsApp community to be the first to receive breaking and exclusive F1 news.
According to The Sun, Sir Dave Brailsford is set to become a senior figure in United’s new-look sporting hierarchy and could approach Potter if Ten Hag’s position comes under scrutiny.
The INEOS director of sport is said to be a huge admirer of Potter and was keen to appoint him as Nice manager.
However, the report adds that the 48-year-old Englishman could soon be off the market amid interest from Serie A champions Napoli.
They are claimed to be keen on speaking to Potter about replacing Rudi Garcia in the dugout, with Antonio Conte also in the frame over a return to Italy.
Potter is still highly rated for his successful spells at Swansea and Brighton and Hove Albion despite his disastrous eight-month tenure as Chelsea head coach.
A tumultuous reign under the new ownership of free-spending Todd Boehly featured 11 defeats in his 31 games in charge, achieving the lowest points-per-game return of any Chelsea manager in history.
Potter has been out of work since his sacking in April when Chelsea were languishing in 11th in the Premier League with 28 points from a possible 66.
But he still has his suitors in high places, including at Old Trafford once Sir Jim and his INEOS team come into power as early as this year.