The Turkish side bought Szymanski in the summer as a replacement for Arda Guler, who they sold to Real Madrid.
He has since become a threat in front of goal for Fenerbahce, making 19 goal contributions in 25 matches overall and showcasing his quality in the Turkish Super Lig.
Szymanski is now being linked with a move away just six months after his move to Turkey and, according to Turkish daily Star, Klopp has been tracking him for four years.
The German is a big fan of the Poland midfielder and could look to take him to Anfield.
However, Fenerbahce are clear in their view that they do not want to let Szymanski leave Istanbul in January.
The Turkish giants are claimed to rate Szymanski at over €20m.
Affording Szymanski is unlikely to be a problem for Liverpool, but whether Fenerbahce can be convinced to sell in January remains to be seen.