According to the Guardian, United are 'preparing an opening offer' for Sessegnon worth around £25m, having had scouts monitor the 17-year-old for a number of months.
It is said that manager Jose Mourinho would be willing to let Shaw leave the club should Sessegnon arrive, with Tottenham tipped to reignite their interest in the former Southampton starlet if that is the case.
Spurs have also been linked with Sessegnon themselves, while even Real Madrid are said to have been watching. He is capable of playing as a conventional left-back, wing-back, or even higher up the field - he recently scored a hat-trick in a Championship game against Sheffield United.
Mourinho will have been well aware of Sessegnon's talents for some time, even prior to his emergence at first-team level in 2016, with his son, Jose Jr., formerly part of Fulham's youth setup and in the same age group as Sessegnon.
United's left-back situation is somewhat bizarre at this moment in time. Mourinho has a number of options but doesn't seem to fancy Matteo Darmian or Daley Blind - the latter has curiously only played Champions League games since September.
Veteran utility star Ashley Young has been getting the nod from the manager, but Shaw has also impressed when he has been given rare opportunities in recent weeks. Marcos Rojo is another option, but Mourinho has generally preferred the Argentine as a centre-back.