Asllani, 23, is expected to put pen to paper on a contract with Leipzig before the end of the month, with the report suggesting the transfer could be finalised within the next two weeks.
His existing deal at Hoffenheim runs until 2029, but the release clause has made him accessible to any club willing to meet the fixed fee, and RB Leipzig have moved decisively to beat the competition to his signature.
Both Barcelona and Borussia Dortmund had also been credited with interest in the Kosovo international, but neither move gained serious traction. Barcelona were only ever considering Asllani as a fallback option behind their primary target, Atletico Madrid's Julian Alvarez, while Dortmund's interest was complicated by the need to first create space in their squad and wage bill, with the club's long-mooted sale of Serhou Guirassy having stalled for some time.
Asllani finished last season with 10 goals and seven assists for Hoffenheim, form that has established him as one of the Bundesliga's more sought-after attacking players heading into the new campaign.
His move to Leipzig would represent a switch between two established Bundesliga clubs, with the Saxony side looking to strengthen their forward options as they aim to build on last season's performance in Europe.
Hoffenheim's sporting director, Andreas Schicker, had already confirmed publicly that the club was in talks with domestic and international suitors over Asllani's future, acknowledging that the release clause made a departure a realistic possibility this summer.
The German club will now look to reinvest the fee generated by his sale as they continue to rebuild their own squad following a difficult recent campaign.
For Leipzig, the addition of Asllani would bolster an attacking department that has faced added uncertainty this summer amid ongoing questions over the futures of other forwards at the club, giving new head coach Martin Demichelis a fresh option to call upon as preparations for the new season continue.