Chairman Massimo Cellino is keen to avoid losing Byram on a free in the summer, even though they would be eligible for compensation through a tribunal due to his age.
He said: "The fans might not know this but if Sam doesn’t sign a contract then someone in the summer still has to pay about £2m for him.
"It’s based on earnings over the last four years and Sam has earned a lot of money in that time. It is also based on appearances and potential.
"If you’re in the Premier League and thinking he’s going to cost £5m to £7m in January, do you wait until the summer when there’ll be an auction of three or four clubs looking to get him under the compensation scheme? You would go get him now, wouldn’t you?"
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