Wilson has been told he can leave Old Trafford on loan by boss Jose Mourinho.
He spent most of last season with Brighton, scoring five goals in 27 appearances for the Championship club.
Derby hope the opportunity of regular first-team football will appeal to the 20-year-old, who featured twice for United last season.
The Rams have only scored one goal in their three league games so far this season.
Last season's top scorer Chris Martin was the subject of a £6.5m bid from Wolves earlier this month, which was rejected.
Find all the latest football transfers on our dedicated page.
Subscribe to the BBC Sport newsletter to get our pick of news, features and video sent to your inbox.
Championship clubs Derby County and Fulham are among those to have asked about taking Manchester United striker James Wilson on loan.