The Galatasaray forward was reportedly a target for West Ham and Tottenham while playing for Trabzonspor last season, but instead opted to stay in Turkey with Galatasaray.
The 27-year-old hit 33 goals for Trabzonspor last year, and already has five goals in his first six league appearances this campaign.
Yilmaz is valued at £13m, and could represent good value for a Liverpool side keen to add an out-and-out poacher to its squad.
Brendan Rodgers offloaded Andy Carroll at the end of the summer transfer window, and failed to secure a replacement, leaving the Reds with Luis Suarez and Fabio Borini as their only recognised senior strikers.
Liverpool have been linked with a number of targets, including Athletic Bilbao hitman Fernando Llorente and Crystal Palace starlet Wilfried Zaha, but talkSPORT reports a bid for Yilmaz could be on the cards.
The Turkey international has played for all four of the big clubs in his homeland, including Besiktas, Fenerbahce, Trabzonspor and most recently Galatasaray.
But Yilmaz was never considered prolific before his 2010 move to Trabzonspor. Since then, however, the Turkish star can't stop scoring, and could be a good bet to lead the Liverpool attack.