City are believed to have offered the asking price of around 1m euros to Bursaspor, twice the offer that was thrown out from Rangers.
"Bursaspor have accepted a bid and we're trying to get it done," the Mackay said.
"I've explained my thoughts on Cardiff and what we can offer him.
"He's a top footballer and getting better and better.
"I can only control what I can control and that's my club. If other clubs make bids, that's up to them."
Miller is expected to undergo a medical in Cardiff later today.
