Bild is reporting that the 23-year-old has opened discussions with the Bundesliga third-placed side despite competition from Borussia Dortmund and Barcelona.
Asllani is available via a €30m release clause in his Hoffenheim contract — a price that makes the operation financially straightforward for any club prepared to commit. He contributed 19 scoring involvements across all competitions last season, establishing himself as one of the most versatile attacking midfielders in the Bundesliga with the ability to play centrally, behind a striker or on the wings.
RB Leipzig's interest is at an early stage. The club already have three centre-forwards under contract — Romulo, Conrad Harder and teenage talent Samba Konaté — and their wider attacking positions are well-stocked. The talks with Asllani reflect an openness to adding quality rather than an urgent positional need, and no decision to intensify negotiations has been taken.
Dortmund are the more advanced pursuer. They have been tracking Asllani for several months, attracted by the flexibility he would offer across their attacking line. However, BVB face the complication of Serhou Guirassy's uncertain future — and that situation must be resolved before their own recruitment can be properly structured.
Barcelona are also understood to retain interest. Whether any of the three clubs triggers the clause before the summer window closes will depend in part on the World Cup — Asllani is part of Kosovo's squad — and what emerges from each club's own outgoing business in the weeks ahead.