The Polish striker earns £2million a year, almost £40,000 a week, at Dortmund.
But United would be prepared to pay him £90,000 a week — nearly £4.7m a year, according to British Daily Tabloid, The Sun.
The player was expecting to follow his team-mate Mario Gotze to Bayern Munich.
But Dortmund blocked that this week saying they were adamant he would not go to their Bundesliga rivals and the team that beat them in the Champions League Final.
The player himself was stunned by the news, but does not believe he will still be at Dortmund next season.
Lewandowski said: “I do not think that this is the final decision and I expect the situation to change.
“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.”
United have now been told they can have the Polish striker if they offer the German club£25.5m.
Dortmund chief executive Hans-Joachim Watzke has said that the deal to Munich will not go through because they simply had not bid.
Watzke said: “Robert will definitely not transfer to Bayern Munich in 2013.
“This is final. We have informed Robert and his advisers. The Bayern case is closed because there was no written offer.”