United paid a world-record transfer fee for a defender when they signed Harry Maguire from the Foxes for £80m this summer.
And Ole Gunnar Solskjaer is also a huge fan of Maddison, who has made an excellent start to the new season.
The former Norwich attacking midfielder is set to be named in Gareth Southgate's England squad for the September international break.
United will attempt to sign Maddison in January, although they realise it will be difficult to prise him away from the King Power Stadium in the middle of the campaign.
The Red Devils are also aware that Maddison will not come cheap, with Leicester likely to demand around £80m.
Solskjaer's side suffered their first defeat of 2019/20 on Saturday, losing 2-1 at home to Crystal Palace.
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