Sky in Germany claim Liverpool are assessing the possibility of a move, as Arne Slot looks to inject fresh spark into his attacking options.
Gnabry, now 30, joined Bayern in 2017 after a spell with Arsenal and has since become one of the Bundesliga club's most reliable performers. He has amassed 298 appearances and scored 97 goals in all competitions during his time in Bavaria.
However, his future remains uncertain, with his current deal due to expire at the end of the season. From January, he will be free to negotiate a Bosman transfer with foreign clubs.
Bayern are understood to be keen to avoid losing him for nothing and are continuing talks over an extension, though no agreement has been reached.
Liverpool, meanwhile, see an experienced and tricky wide player as a priority target, particularly given their recent dip in form.
Slot's side head into the international break after back-to-back defeats, including a bruising 3-0 loss to Manchester City that followed a 2-1 setback at Crystal Palace in September.
The downturn has stalled what had been a promising run, which featured consecutive victories over Aston Villa and Real Madrid.
After 11 league matches, Liverpool sit eighth, eight points adrift of leaders Arsenal, prompting growing questions from pundits and supporters about how Slot can restore momentum.
