In addition to the £25.5million that the Champions League finalists are demanding for his signature, the Premier League champions might have to stump up extra cash to persuade Bayern to rip up their agreement with Lewandowski.
The 24-year-old striker has attracted widespread interest over the past few months, after a standout season for Jurgen Klopp's side in 2012/13.
Last month it was reported that Lewandowski would follow in the footsteps of Mario Gotze and complete a controversial move to the Allianz Arena this summer.
However, Borussia Dortmund are defiantly refusing to deal with their Bundesliga rivals, paving way for Manchester United to hijack the transfer.
New Red Devils boss David Moyes is looking to exploit the uncertain situation, but according to sources in Germany will have to pay Bayern Munich compensation in order for Lewandowski to complete a switch to Old Trafford.
Pep Guardiola - who takes control of first-team affairs from Jupp Heynckes on July 1, will have to make a decision on whether to accept Manchester United's compensation package and lose out on one of Europe's most in-demand strikers, or alternatively hold out for Lewandowski when his contract at the Westfalenstadion expires in 12 months time.
That said, Borussia Dortmund have not given up hope of retaining the player's services and are ready to offer him a new contract that will see the Poland international treble his £20,000-a-week wages.