The Poland international currently earns around £40,000-a-week in the Bundesliga, but Manchester United are willing to offer him a pay packet worth close to £90,000-a-week.
Lewandowski had been strongly linked with a move to Bayern Munich - his Dortmund teammate Mario Gotze has already agreed to join the German champions this summer.
But Dortmund have insisted they will not sell Lewandowski to their Bundesliga rivals, opening the door for United to make their move for the striker.
The report claims United have been told they can sign the Polish forward for £25.5m. But Dortmund are running out of time if they want to recoup a fee for Lewandowski - the forward has just 12 months remaining on his contract and could leave for free next summer.
And Lewandowski has revealed he still expects to leave sooner rather than later despite Dortmund's refusal to sanction a Bayern Munich transfer.
"I do not think that this is the final decision and I expect the situation to change," said the Dortmund star. "I am enjoying my holiday for now but I hope things move forward pretty soon and the situation changes.
"I do not get upset about anything."
Lewandowski has spearheaded the Dortmund attack to two Bundesliga title wins and led the line as Jurgen Klopp's side reached this year's Champions League final.News Now - Sport News