The striker's existing deal is set to expire in 2020 after originally joining the side from Stuttgart back in 2016.
Since his arrival at Red Bull Arena Leipzig, Werner has risen to the top of German football, bagging 60 goals in 108 games for the club.
Leipzig want to keep the striker on their books, but they confirmed on Saturday that he is likely to leave the side in the near future.
Speaking with Sky Germany at half-time during Leipzig's clash with Wolfsburg, CEO Mintzlaff said: "We want to extend it with him, but he has signalled to us that he would rather not extend."
Werner scored in the game against Wolfsburg, helping Ralf Rangnick's side cruise to a 2-0 victory.
Rangnick himself has also spoken of Werner's future in the past , insisting he was hopeful he'd remain at the side for "sporting reasons".
"We have gone to our financial limit with our offer," he said. "We know that he can earn more elsewhere, but he can also provide for himself for the rest of his life here.
"The financial aspect will not be the decisive factor, but rather the sporting perspective.
"We're trying to keep him. He knows he's very popular in the team, with the coach and the fans. He's a top player, and he has become one of the most exciting Bundesliga strikers in the country.
"He still has one-and-a-half years left of his contract, and I hope that he'll stay. But in the end, it's up to him."
Werner has 14 goals and five assists to his name in the league this season, with the club currently third in the division and on course to secure Champions League football.