The midfielder could become available in the market before the transfer deadline on October 5.
Frank Lampard still wants to lure Declan Rice back to Stamford Bridge from West Ham.
And 29-year-old Kante, who suffered with injuries last season, could be sacrificed to raise further funds.
Chelsea have spent big this summer, and while they look to balance the books somewhat, a £60m price-tag has reportedly been put on France international Kante.
United are focusing their attention on strengthening their attack this summer, with Jadon Sancho still their No.1 target.
But Ed Woodward has shown with the £38m purchase of Donny van de Beek from Ajax that they are willing to spend in other departments if the deal is right.
Kante is admired by Ole Gunnar Solskjaer and the United manager is said to know he still ideally needs his own Casemiro-style player at Old Trafford.
Nemanja Matic, who played alongside Kante at Chelsea, is 32 now and unlikely to play more than 50 games in a season.
As a result, the Daily Mail say United would be interested in bringing Kante to Manchester if Chelsea gave them encouragement to complete the signing.
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Inter Milan are very keen on reuniting Kante with former Chelsea manager Antonio Conte, though.
And whether the Blues would genuinely be willing to help out a direct rival in United is another matter.
In the meantime, Solskjaer has a decision to make with regards to his team ahead of the Premier League season starting imminently.
Matic, Bruno Fernandes, Paul Pogba and Van de Week will all be vying for a spot in the side.