The Daily Star says United have agreed a £16m deal for the hotshot and will complete the deal after the tournament.
The Old Trafford outfit snapped up the Mexico striker just before the 2010 World Cup when he went on to make a massive impact.
United are believed to have offered Lewandowski a four-year deal worth around £70,000 a week.
Fergie watched the 23-year-old hitman in action several times last season and wants him to link up with Wayne Rooney.
Fergie has already raided Dortmund for Japan sensation Shinji Kagawa, who has cost United £16m.
It is understood United bosses secured a similar deal for Lewandowski during negotiations with the Bundesliga champions for Kagawa.