The 22-year-old striker is one of the hottest talents in the Bundesliga and has already netted 11 times in the league this season – only Paco Alcacer and Luka Jovic have netted more.
Leipzig are desperate to keep the Germany international, but he has so far stalled on signing a new contract with his current deal expiring in 2020.
According to Bild, Werner is hesitating and believes he is ready to make the next step in his career, with Leipzig only fourth in the Bundesliga and 11 points behind leaders Dortmund.
Liverpool, who have long-held interest in Werner, are watching the developments closely and believe they could land the forward for just £36m if he refuses to renew.