The Poland star has indicated to friends he has set his sights on the move from Borussia Dortmund at the end of the season.
After rejecting overtures from United last summer, the German club have reluctantly accepted they are about to lose their leading marksman.
With his Dortmund deal running out at the end of next season, United hope the fee will be little more than the initial £12million they paid for his former team-mate Shinji Kagawa last summer.
Sir Alex Ferguson believes pairing 24-year-old Lewandowski with Robin van Persie — and Wayne Rooney operating just behind — would give him the most lethal attack in the Premier League.
As well as picking up Lewandowski, Ferguson has told Rooney he needs to get back towards the top of the scorers charts.
The Englishman has only three in 14 games for United, one of which was a penalty.