Sir Alex Ferguson scouted the forward last season while also watching Shinji Kagawa, and the United boss is reportedly still keen on the striker despite signing Robin Van Persie in a £24m summer deal.
Lewandowski is Dortmund's top scorer this season with 19 goals in 27 games, but the reigning Bundesliga champions have struggled to keep pace with table-toppers Bayern Munich.
And Lewandowski's agent, Cezary Kucharski, has dropped the biggest hint yet that the forward is set to leave the German club at the end of the season.
"Anyone who has followed Robert's career will know that he has never played at any club for longer than two years," said Kucharski.
"At Dortmund, he has been playing his third season. Anyone who can read between the lines knows what that means.
The 24-year-old has just 16 months left on his Dortmund contract, and the report claims he could be available for as little as £18m. Manchester United have a number of striking options, including Danny Welbeck, Javier Hernandez, Wayne Rooney and Van Persie, but that hasn't stopped speculation linking the Premier League leaders with the forward.
Bayern Munich have also been linked with the striker - providing Lewandowksi an option should he wish to remain in the Bundesliga.
But Kucharski has claimed that the Polish star has already made a decision on his future, and is simply waiting for the right time to make the announcement.
"Robert has made his decision," Kucharski said. "He alone determines the moment when he will announce it."