The England Under-21 left back has been linked with United and Tottenham in recent months.
The Times says United boss Jose Mourinho is keen on signing at least one full-back in the summer and the club have been monitoring Sessegnon all season.
The 17-year-old won the Championship player of the year and young player of the year awards this month, scoring 15 goals in 44 Championship games.
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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.