Blues boss Alex McLeish is keen to add some quality to his squad in January and has gone after the Scotland international who has scored 21 goals in the Scottish Premier League already this season.
McLeish confirmed an offer had been made but said he was no closer to knowing whether the 31-year old would land at St Andrews or not.
"If there is an opportunity for us to go forward with someone and it is good for the future of the club, then (Blues owner) Carson Yeung likes that kind of strategy," he said.
"We've spoken to Kenny Miller's people and they haven't said yes or no. We'll see where it goes from here."