The Red Devils were short of quality at the back last season, with players like Jonny Evans and Phil Jones not looking at their best as the club dealt with the departures of experienced club legends like Rio Ferdinand and Nemanja Vidic.
With only Marcos Rojo brought in to the heart of the backline, it is clear that the club need more depth in that department, and Stones is one of many central defensive players to be linked with a move to Old Trafford this summer.
Last month, it was reported that United had made a bid for Real Madrid defender Sergio Ramos, according to Sky Sports, but it may now be that Stones is their priority target as they look to get one over rivals Chelsea.
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The Daily Mail have reported that Everton would reject a £30m bid from Chelsea this week, but ESPN claim United sources are tipping them to try their luck with a swap deal with the Toffees, which would likely see Evans move to Goodison Park as part of the deal.
The prospect of a player joining Everton in return could be more appealing to the Merseyside club, though it is also understandable that they want to keep hold of a top prospect like Stones, who is already capped for England and who looks set for a bright future in the game after some superb performances in the Premier League so far in his short career.
Without receiving compensation for the defender, the Toffees may be hesitant to offload Stones, as they are currently short of centre-backs, having lost Sylvain Distin and Antolin Alcaraz.