It was confirmed last week that the Poland international would be joining the Bavarians on a free transfer, after agreeing a five-year contract with Pep Guardiola's side.
Having already lost Mario Gotze to Bayern last summer, Dortmund fans reacted angrily to the news and, for that reason, Lewandowski is entertaining the idea of extra security for both he and his family.
"We are giving serious thought to it," Maik Barthel, one of the forward's consultants, told SPORT BILD.
According to the paper, Lewandowski, whose Facebook profile page has been targeted by irate Dortmund supporters, is considering personal protection after seeing how Gotze was treated when it was announced that he would be joining Bayern.
Indeed, the Germany international employed bodyguards to protect his home and accompany him on his first trip to Die Schwarzgelben's training ground after news of his transfer broke.