The Clarets have agreed to pay £100,000 up front for the Scotland international right-back, with a further £100,000 payable next summer.
"I have known Graham for a few years and we would have liked to have had him five or ten years ago," manager Steve Cotterill said.
"But we have got to the stage now where everything has been agreed and, so long as he passes the medical, we've got him and we are going to enjoy him.
"I don't think there is a question over his quality and his age. He could be one of those players who has got even better with age."