Liverpool have been strongly linked for months with a move for the 24-year-old, but speculation has risen since his return to Bundesliga action in the past few weeks.
It emerged recently that Werner had held talks with Liverpool, who were said to be favourites, but it was revealed earlier this week that Jurgen Klopp’s side are unwilling to meet his £50 million release clause.
Instead, the Daily Mirror now reports that the Anfield club will call off their pursuit of the striker unless Leipzig agree to a significant price drop, which is unlikely.
Indeed, Leipzig CEO Oliver Mintzlaff told Bild in Germany: "(Werner's price) won't get any cheaper. We will not sell a player below value if he is under contract for more than a year".