The 18-year-old made only seven substitute appearances in the Premier League last season.
Fulham had included a purchase option in their loan agreement with Bayern, but did not exercise it.
The London club still want to keep Kusi-Asare, though at a reduced price, and have now tabled a bid worth €3m.
Bayern have rejected the offer, demanding a higher fee for the young striker. According to Bild's "Bayern Insider" podcast, Fulham are expected to return with an improved proposal.
Kusi-Asare remains under contract with Bayern Munich until the summer of 2027, giving the German club leverage in negotiations even after a season in which the player struggled for regular minutes in England.
The teenager arrived at Fulham with high expectations after impressing for Bayern's academy sides, but found first-team opportunities difficult to come by under head coach Marco Silva, with his appearances largely restricted to substitute roles in the closing stages of matches.
Bayern's reluctance to lower their asking price suggests the club still rate Kusi-Asare's long-term potential highly, even if his first season in England did not produce the breakthrough either party had hoped for when the loan was agreed.
Whether Fulham's improved bid is enough to close the gap with Bayern's valuation will likely determine whether the Swede returns to Craven Cottage for a second season, continues his development elsewhere on loan, or returns to Bavaria to fight for a place in Bayern's first-team squad.
For now, the saga continues to play out at a relatively modest financial level compared to many of the bigger transfer sagas of the summer, but it remains a meaningful decision point in the early career of a player Sweden's youth set-up have long considered one of their most promising attacking prospects.