The Clarets have been showing keen interest in signing Vincent Kompany’s countryman and a deal appeared to be on the cards.
AC Milan want to offload Origi, who is on a lucrative contract at the San Siro but has failed to deliver since joining from Liverpool last summer.
If he does leave AC Milan though it will not be to Burnley, with Origi rejecting the switch to Turf Moor.
It is unclear why the striker is not sold on a return to the Premier League with Burnley, but he does not want the move and it will not happen.
Burnley remain in the market for additions before the transfer window slams shut on Friday night.
Origi has not yet played for AC Milan this season and the Italians are hopeful they will still be able to shift him out.
He scored just twice in 27 Serie A outings for the Rossoneri last term.
