It was claimed by the Daily Mail last week that the Clarets, along with Brighton and Sheffield United, were considering a move for the 25-year-old.
However, as reported by 90min, that story has since developed. A summer deal is slightly complicated but not entirely impossible.
Hendry is on loan at Belgian club KV Oostende, who have an option to buy the Scotland international at the end of the season for around £2million.
Effectively, Oostende can then sell for a quick profit with £10m the rumoured asking price – money Celtic would not see.
It is claimed Aston Villa, Brighton and Wolves are keen while Newcastle, Leicester, West Ham and Crystal Palace have also kept tabs on his progress.
The report adds that Burnley apparently had a bid rejected for Hendry in January after Oostende were unwilling to cut short his season-long loan.
Oostende are fully aware of the emerging interest. In an ideal world, they would prefer to keep him but are open to the idea of cashing in immediately.
Meanwhile, former Clarets goalkeeper Paul Robinson believes Robbie Brady would jump at the opportunity to join Celtic.
Brady has been regularly linked with the Hoops ahead of his contract expiring in the summer.
“I think he would be a great signing for Celtic,” Robinson told Football Insider, as quoted by Pundit Arena.
“He is a fantastic player, a hard-working winger who has a fantastic delivery on him.
“I know it is a move that he would really love as well with his Irish background. I think the chance of playing at Celtic would be a big draw for him and one that would be difficult to turn down.
“He has had his injury problems this season and he has been in and out of the team. To let his contract run this far down tells me that he is probably leaving.
“A move to Celtic would work. There is going to be an overhaul there this summer but I think Robbie would be a great addition regardless of who comes in.”
Here are the latest transfer headlines from Burnley and their Premier League rivals: