The Sun says Lewy will be offered a contract worth £80,000 a week over five years.
That £20m jackpot doubles the money the Pole, 24, is paid by the German champs.
It will also be seen as a huge statement of intent by United boss Alex Ferguson.
His side have already netted 48 League goals in 19 games this season — 11 more than second-highest scorers Chelsea.
Dortmund had to fend off interest from United last summer, after Lewandowski hit 30 goals to lead them to a German League and Cup double.
The player’s contract is up at the end of next season.