Gundogan was influential in his side's 4-3 aggregate win over Real Madrid to reach the Champions League final.
Manchester City were originally linked with the £17m rated player, but their interest appears to have cooled.
United are intent on revamping their midfield, as Paul Scholes is in the autumn of his career and Nani and Anderson are tipped to leave Old Trafford.
Gundogan was coy about his Dortmund future recently, telling German paper Sport Bild: "It is not reprehensible to answer such questions (of a future career in England or Spain) with 'yes'.
"Everyone has a childhood dream on what he would like to achieve one day and my dream always was to play in England or Spain."