Barton’s QPR future remains a doubt after his sending-off in the final game of last season against his former club Manchester City, for which he received a 12-match ban for his actions following his dismissal.
Rangers’ boss Mark Hughes also already stripped the 29-year-old of the club captaincy and it is believed he is willing to listen to offers for his controversial midfielder.
Owls’ boss Jones is trying to build a side to compete in the Championship and admits he would be keen on bringing Barton to Hillsborough and would not be put off by his reputation.
"If he did become available it's something we would look at, but there would be many things to consider," Jones told The Sun.
"I wouldn't be frightened of bringing him here if he became available.
"His reputation would not put me off."
Barton moved to Loftus Road from Newcastle at the beginning of last season for £5.8m but could see his first team opportunities limited after QPR signed Park Ji-Sung and Samba Diakite and were linked with a move for Blackburn's Junior Hoillett.