The Whites have already paid RB Leipzig £15.5m over the saga, bringing the total cost to £40m for 48 minutes of football.
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Leeds have since appealed the decision which was put forward by FIFA's Dispute Resolution Chamber.
The club insist they do not need to pay the transfer fee, given the players' loan expired on June 30.
Leeds also claim they were promoted when the campaign was delayed due to the COVID pandemic.
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Augustin joined the Whites from RB Leipzig on-loan in January 2020, with an obligation to buy for £18m.
The 25-year-old made the permanent switch to Elland Road after Leeds' Premier League promotion, but played just 48 minutes of football.
Leeds managed to reach an agreement with Leipzig, paying a £15.5m settlement, with the player now featuring for Swiss side Basel.
The £24.5m fee needed to be paid is the full five-years of his contract, having been on wages of around £80,000-a-week.
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A Leeds spokesman has since stated: "The club cannot comment on an ongoing legal matter."
talkSPORT's Alex Crook weighed on the situation involving Leeds and Augustin.
"They [Leeds] are going to appeal it, but the whole issue surrounding the technicality, basically they are trying to use the pandemic as an excuse of not to pay what they agreed to.
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"So, I don't have a lot of sympathy for Leeds in this case, because they were promoted to the Premier League.
"Ok [Jean-Kevin] Augustin did not play a too big of part in that, but you can't agree to sign a player subject to Premier League promotion, and effectively decide we are going to renagon that, because he is not good enough."
"I don't think Leeds have a leg to stand on."