The Scottish champions accepted an offer in the region of £600,000 from Blues after the 31-year-old failed to agree a new deal to remain at Ibrox.
Miller, whose current contract is set to expire at the end of the season, had also been holding talks with Italian outfit Fiorentina but could be set for a move to Turkey instead after apparently accepting an "irresistible" offer from Bursaspor.
The forward, who has netted 22 goals for Rangers so far this term, was first linked with the Turks after facing them in the Champions League this season. The news of his possible departure was revealed on Blues' official website, when they confirmed they had failed in their attempts to lure Miller to the Midlands.
A statement read: "Kenny Miller will not be joining Blues after the player's representatives informed the club that he has accepted an irresistible offer from Turkish champions Bursaspor. The club pulled out all the stops in an attempt to bring the Scotland international striker, who attracted offers from several top European clubs, to St Andrew's.
"However the offer from Bursaspor proved to be too good for Miller to turn down and the 31-year-old has now decided to undertake the next chapter of his career in the Turkish Super Lig.
"The Glasgow Rangers man was said to have been torn over the last seven days as he considered his future and the decision to turn down the chance to link up with Alex McLeish again came as a very difficult one.
"Despite this unavoidable set back, Blues fans can rest assured that the Board and management team are continuing in their efforts to further complement the playing squad."