Bayern are set to host Zinedine Zidane's side in the semi-finals of the Champions League on Wednesday, with speculation rife that Lewandowski could be lining up for the Blancos come next season.
Madrid are set to bolster their forward line this summer with Karim Benzema struggling in front of goal and Gareth Bale unable to make any impact following a spate of injuries.
Lewandowski is seen as the ideal addition, with the 29-year-old having scored 39 goals in 43 appearances across all competitions for the Bundesliga champions this term.
The Poland international is now keen to make one final big move before entering the twilight of his career, and earlier this season switched from his agent of ten years, Cezary Kucharski, to 'super-agent' Pini Zahavi, who has previously represented Rio Ferdinand, Carlos Tevez and Javier Mascherano.
His current deal at Bayern does not run out until the summer of 2021, however, and the Bundesliga champions are set to dig their heels in over one of their prized assets.
Given their status as one of the biggest clubs in European football, Goal understands the Bavarians do not feel they can be forced into selling any player whom they want to keep hold of while also wanting to give incoming manager Niko Kovac the best possible chance of success.
Madrid had been hopeful that the ease in which an agreement was reached for James Rodriguez to move to Germany would help them in negotiations for Lewandowski, but it seems their pursuit of the former Borussia Dortmund man will not be as straightforward.
This is not the first time the two clubs have found themselves in this kind of situation, with Bayern having previously refused to sell Franck Ribery in 2009 despite having picked up Arjen Robben from Santiago Bernabeu earlier that year.