The 22-year-old is being monitored by the German champions, with the Bild podcast Bayern Insider reporting that the club see significant long-term potential in the attacking midfielder.
The assessment is understood to be that Gruda has the qualities to develop into a Bayern player, although the club are not currently negotiating a transfer with RB Leipzig.
Gruda joined RB Leipzig from Brighton, with a purchase obligation in his loan agreement set to be triggered once he reaches a specified number of appearances. Leipzig are therefore expected to pay €25 million to Brighton next summer.
The Germany international prospect has made an encouraging impression this season, operating primarily on the right wing but also capable of playing centrally. His creativity and ability to influence attacking situations have helped him attract attention from the national team setup.
Gruda has been called up by Germany coach Jürgen Klopp for the forthcoming international fixtures and could make his senior debut.
Bayern's interest is viewed as a longer-term possibility rather than an immediate transfer plan. There have been no negotiations between the Munich club and Leipzig, but the Bundesliga leaders are continuing to assess his development.
If Gruda maintains his current form and establishes himself in the Germany squad, Leipzig are expected to place a substantial valuation on him. His performances over the coming months could therefore have a significant bearing on his future.