The 25-year-old is moving across Germany to the European champions on a free transfer when his Westfalenstadion contract expires in the summer.
Although Bayern were always strong favourites to secure his signature, several clubs had been linked with a swoop for the Poland international and Lewandowski has admitted to exploring other options.
"The only thing I can say is that Florentino spoke with me," he told a press conference on Tuesday. "It was a very brief conversation, it's the only thing I can say about the meeting with the president of Real Madrid."
Lewandowski played for Dortmund against Madrid on Tuesday as they nearly produced a remarkable three-goal comeback to progress to the Champions League semi-finals.