Spox credited both clubs with interest in Knasmüllner late last week, however, its latest report includes no mention of Wednesday.
Instead, it claims an agreement has been reached between the 12-goal Admira Wacker player - who is out of contract at the end of the campaign and can therefore talk to other clubs about a summer free transfer - and Barnsley.
Barnsley, though, appear to want to bring forward Knasmüllner's move. According to Spox, however, the Tykes' offer falls short of Admira's €500,000 (£440,000) asking price - and both parties 'seem reluctant to give in'.
It remains to be seen whether or not Sheffield Wednesday - who recently signed Dutch midfielder Joey Pelupessy in similar circumstances - will respond to the alleged development.
New Owls manager Jos Luhukay may be familiar with Knasmüllner, who was on the books of FC Ingolstadt 04 during the Dutchman's spell in charge of fellow German sides FC Augsburg and Hertha BSC.
Can Wednesday still beat Barnsley to Knasmüllner's signing?