Blues have also made a bid for out-of-favour Tottenham striker Robbie Keane in the last few days.
Miller, 31, had previously played south of the border for Wolves and Derby and has been a consistent scorer throughout his career. He rejoined Rangers from the Rams for a second spell during the summer of 2008.
Miller, who has also had spells with Hibernian and Celtic, has won more than 50 international caps during the past decade.
McLeish is aware his side have to start being more clinical in front of goal, with midfielder Craig Gardner their leading scorer this season with just five goals.
If McLeish gets his man, Miller could make his debut in the derby clash with Aston Villa at St. Andrew's in nine days time.
