The Bluebirds have had a bid accepted by the Turkish club and are now discussing personal terms with the 31-year-old.
"Bursaspor have accepted a bid and we're trying to get it done," said manager Malky Mackay. "He's a top footballer and getting better and better."
Former club Rangers tried to re-sign the Scotland international but had a bid of £400,000 rejected. It is thought that the Welsh club have offered more than double that figure for his services.
Despite moving to Turkey in January, Miller remained the SPL's top scorer last season with 21 goals.