This is according to talkSPORT, who understand United will face stiff competition from Arsenal for the world-class 25-year-old who has scored 11 goals in 24 appearances for BVB this season.
But the report adds that Real Madrid, according to Spanish newspaper AS, could return with a bid for Reus in the future. The Spanish giants are not interested in the winger at this moment in time on account of possessing a host of stars who play in similar positions.
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Real Madrid snubbed the opportunity to sign Reus last summer, instead opting to sign Colombia international James Rodriguez, who has scored an impressive 15 goals and provided 17 assists since arriving from AS Monaco.
But despite finishing the season without a major trophy - aside from the UEFA Super Cup and FIFA Club World Cup - Real Madrid are keen to stick with what they have in terms of attacking personnel.
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Bale and Ronaldo to stay at Real Madrid
Rafael Benitez, the manager expected to replace Carlo Ancelotti at the Bernabeu next week, will be expected to get the best out of the likes of Gareth Bale, Isco, Toni Kroos, Jese and the likes next term - as well as ensuring Cristiano Ronaldo continues to score a goal every game.
Karim Benzema has been linked with a move away from Real Madrid, and if the Frenchman leaves this will free up room for either Ronaldo or Bale to become the team's first-choice centre-forward.
It seems very unlikely that Ronaldo or Bale will end up at Manchester United this summer - and this could tempt the Red Devils to sign Reus instead.
Reus's recent injury record is a concern
Louis van Gaal, however, might have concerns about Reus's worrying recent injury record.
The former Borussia Mönchengladbach star missed the 2014 World Cup finals in Brazil after suffering a serious ankle injury prior to the tournament, and missed a significant portion of last season after tearing ligaments in his right ankle while playing for Dortmund in a league match against Paderborn.
With so many of his players missing matches due to injury this term, Van Gaal needs reliable performers who are available for selection on a weekly basis.
But if Reus can steer clear of injuries, there are few better attacking players in world football.
This is a player who would make the difference for either Manchester United or Arsenal next season in both the Premier League and the Champions League.
Manchester United fans: do you want Louis van Gaal to sign Marco Reus this summer? Have your say by leaving a comment in the box below!