Spanish media outlet Don Balon say the Portuguese chief is desperate to sign the German forward.
United need to bolster their attacking options this summer.
Wayne Rooney looks set to leave Old Trafford at the end of the season and Reus could fill the void.
He joined Dortmund from Borussia Mochengladbach in 2012.
And since making the switch, Reus has played 123 games for the club and scored 54 goals.
It's suggested the 27-year-old could be willing to leave his homeland.
But a move to Old Trafford is far from guaranteed.
Don Balon say Atletico Madrid are also sniffing around.
Diego Simeone is plotting a summer overhaul after his side finished third in La Liga.
Antoine Griezmann is widely expected to be sold and Reus might be the ideal replacement for the influential Frenchman.
