The 24-year-old Argentine international has been in impressive form for Benfica this season, particularly impressing in the Champions League.
Red Devils boss Sir Alex Ferguson is keen to strengthen his midfield ranks at Old Trafford in the summer, and has identified Gaitan as a possible summer acquisition.
However, Manchester United will now face stiff competition for the South American's signature after the Bundesliga giants' interest was revealed by Portuguese newspaper O'Jogo.
The reigning Premier League champions are rumoured to have already lodged a £20million bid for Gaitan, but Benfica will not accept any offer less than £25million.
Bayern have already proved to be a thorn in the side of United, when the German outfit beat them to the signing of former Basel winger Xherdan Shaqiri, and will now be looking to do the same this summer.