The 24-year-old made a deadline day move to the Bundesliga champions, who welcomed the striker with open arms as he aims to play as a backup to Harry Kane this season.
The arrival of strikers, Joao Pedro and Delap, pushed him out of West London and manager Enzo Maresca made it clear that he would have no place in his plans for the rest of the campaign as The Blues chase down the title.
Now, speaking about his future to BILD, Jackson revealed that he may stay with the German side if his move is deemed a success.
“My focus is on giving my best here. Bayern is one of the best clubs in the world.
“If everything goes well and everyone is happy, I would, of course, like to stay. But now it's about the present. Work hard, do my best.”
The Senegal international joined Bayern for a £14.3M loan fee close with a conditional obligation to buy for £56.2M that is unlikely to be paid by the club as it requires 40 appearances that he is unlikely to make.
Bayern face Chelsea in their opening Champions League clash on Wednesday, and Jackson may get his chance to prove to Maresca that he should not have been sold this summer.