The 22-year-old broke a bone in his hand while training with the Scottish national team on Monday morning.
Burnley boss Brian Laws is hopeful Fletcher will be fit for the clash with Arsenal on Saturday.
“He has been for an x-ray and is coming back to us,” Laws told the club’s website.
“We need to get it checked out again but we feel confident that he will be available to play at the weekend with some sort of cast.”
