According to Sky Sports' Guillem Balague, Los Blancos are serious about signing the Poland international and the player himself is also keen on the move.
Despite suggestions from figures at Bayern that Lewandowski will not be sold, Balague said the European champions are intent on making this one happen:
"More and more details are coming out since he changed his agent. Now we are hearing through German media that he's told his team-mates he wants to go to Real Madrid, all part of the same strategy to get out of Bayern Munich. [Bayern] insist Lewandowski will not be leaving but Real Madrid are prepared to pay a big sum for him."
Balague also reported that both Cristiano Ronaldo and Sergio Ramos have spoken with the striker and encouraged him to make the switch to the Spanish capital.
Lewandowski is one of the best goalscorers in football, and with that in mind, there's no doubt Madrid would have to break the bank to sign him. As the UEFA Champions League Twitter account noted, the 29-year-old on track for his most prolific Bundesliga campaign:
The consistency with which the Bayern No. 9 has found the net down the years has been remarkable—he's scored at least 17 league goals in each campaign since 2011/12.
Lewandowski may not be the most dynamic forward, but his all-around game makes him such a threat. Aside from razor-sharp predatory instincts, the Poland international is strong with his back to goal, is able to link play and works hard for his team at the point of the attack.
Even for a team like Madrid, who boast so much attacking firepower, he would represent an upgrade on their options at centre-forward.
As Jason Pettigrove of Marca noted, the two European giants are set to meet in what will be an intriguing UEFA Champions League semi-final:
Madrid would benefit from a new striker. While Cristiano Ronaldo remains as prolific as ever, Karim Benzema's form has dropped off and Gareth Bale hasn't looked comfortable when shoehorned into the team as a striker.
Per the Spanish Football Podcast, the club appears to be losing patience with the struggling attackers:
Lewandowski has the quality to make an immediate impact for Madrid. Given his experience and the success he's enjoyed at Bayern, the prospect of a switch to the Santiago Bernabeu shouldn't be daunting for him, and the prospect of a new challenge may push his game on to new heights.
Still, there will be doubts about the deal, especially given Madrid would have to pay a "big sum" to land a player who will turn 30 in August. Bayern would also surely be reluctant to part ways with their key attacker.