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The 18-year-old striker's deal expires at the end of the season and Dortmund says it has presented him with a "very attractive offer" to renew.
According to sporting director Sebastian Kehl, the club have now issued Moukoko and his advisor with an ultimatum so the player can "devote himself entirely to the sporting challenges" before the Bundesliga resumes this weekend.
Kehl said: "We have put together a very attractive offer that gives him a lot of room for development. That should lead to a decision. He has to make it with his environment and his advisor.
"Youssoufa can now accept this offer and commit to Borussia Dortmund, or we will go our separate ways. I would like him to choose us because he is not yet at the end of his development and I see huge potential in him. But there are limits for us as a club. And we have shown these limits. Now it's up to him to decide.
“Youssoufa and his advisors know that we are expecting a decision this week because we want to have this clarified before the first game back [after the World Cup break] so that the boy can devote himself entirely to the sporting challenges."
Moukoko – who was part of the Germany squad at the World Cup in Qatar – became the Bundesliga's youngest ever player when he replaced Erling Haaland just a day after his 16th birthday in the 5-2 win against Hertha Berlin in November 2020.
He has scored six goals in 14 games (eight starts) in the Bundesliga this season, but Kehl believes there is plenty more to come from him.
"Youssoufa is a great boy," he said. "He's been with us for almost seven years, we trained and developed him. He became a professional and a national player with us. He received an incredible amount of support from us and worked hard to achieve success himself."
Kehl also said Dortmund are "very relaxed" about Marco Reus and Mats Hummels, who are both also out of contract at the end of the campaign.
"From the conversations with both of them I can say that we are very relaxed about the situation," he added. "Both have already made public statements about this in the past few weeks. Mats has decided that he wants to wait and see how things develop for him. He said that publicly himself.
"Of course the players are concerned about their future. That's normal. I'm convinced that we'll find a way to progress together in talks."