Striker Winnall has hit an impressive 11 goals in 20 appearances in the Championship for the Tykes this season but is out of contract in the summer and has so far refused to enter into talks over a new deal, making it increasingly likely he will move on this month.
Because of his contract situation, he could be available for as little as £500,000 and Rangers, along with Old Firm rivals Celtic, have been tracking his progress.
However, Norwich are now interested too and could follow Sheffield Wednesday in making a bid for the in-demand forward.
Rangers boss Mark Warburton has confirmed he wants to strengthen this month and he's said he's close to completing some deals, with Winnall a potential Ibrox arrival.
The 25-year-old came through the ranks at Wolves before several loan spells and ended up at Scunthorpe before joining Barnsley in summer 2014.