The German winger has long been a target for the Red Devils, who have been closely monitoring him over the last year ahead of what looks set to be a massive off-season in terms of transfer business at Old Trafford.
It is rumoured that United failed with a move for the 24-year-old in January, but now they are ready to tempt the Bundesliga club by offering Kagawa as part of any agreement.
The Japanese playmaker left Dortmund in 2012 to join the current Premier League champions, after impressing in the yellow shirt.
However his form has not translated to the English game, and Moyes is thought to have lost patience with him.
Kagawa is still held in high regard at the Westfalenstadion – where fans even started an online campaign to get him back earlier this season – which could tempt the club to accept a swap deal.
Reus is seen as one of United’s top targets, with skilful wide-man having been in excellent form for Dortmund for some time.