The Red Devils are building a team for next season and beyond.
While Shaw has impressed in recent weeks, The Sun suggests United are in the market for a left-back.
And the paper claims United are willing to pay the £60 million Leicester want for the player, which puts them ahead of Chelsea in the race to sign him.
With young Brandon Williams also in the squad, it could mean United sell Shaw to fund a move for Chilwell.
Shaw has started every United game since the Premier League restart.