The Daily Star says United could have to stump up extra cash to persuade Bayern to rip up their agreement with the striker.
Sources in Germany believe that a complicated deal could be required to capture the forward.
United would first have to pay European champions Bayern a compensation fee. Then they would have to negotiate a deal with Dortmund.
Dortmund want £25.5m for Lewandowski, even though he has just one year left on his contract.