The move should by now have been complete, with Leipzig and Hoffenheim having reached agreement over the transfer of the centre-forward, reportedly built around Asllani's roughly €25m release clause and a contract that would have run until 2031.
Asllani travelled to Salzburg to undergo his medical, only for RB Leipzig's doctors to raise concerns and effectively veto the deal, prompting the club to withdraw.
According to Bild, the medical staff harboured long-term concerns about the striker that a short period of rehabilitation would not have resolved, and it was on that basis that officials decided to call off the transfer of the Kosovo international.
Hoffenheim's own statement referenced "medical concerns" too, though the club's medical department does not share Leipzig's assessment.
Within Hoffenheim, sources cited by Bild describe the Leipzig doctors' concerns as "massively exaggerated." Asllani is now being reintegrated into training at the club, and in purely sporting terms his continued presence represents a boost for Hoffenheim, given he contributed 11 goals and ten assists last season, a return that played a key role in the club's Europa League qualification, having signed a contract extension to 2029 only a year ago.
Because of an exit clause in his contract, Hoffenheim's hands had been tied over letting him leave.
"The fact that this transfer isn't happening now is anything but bad news for us from a sporting perspective," said sporting director Andreas Schicker.
For now, Hoffenheim can count on their striker, though the club has indicated it would not stand in his way again should another interested club emerge and want to seek a third medical opinion of their own.
Both Borussia Dortmund and Bayern Munich have previously been credited with interest in Asllani this summer, and the collapse of the Leipzig deal could yet reopen that competition.
Leipzig, meanwhile, are exploring alternatives on the transfer market after also losing Yan Diomandé to Real Madrid this summer, leaving the club short of options up front once again.