Mills, 27, spent last season on loan at The Riverside, helping the Teessiders win the Carling Cup and a place in the UEFA Cup.
But his current Leeds deal has another four years to run and they refuse to match his reported £40k-a-week wages.
However, Birmingham City are ready to match his wage demands and are optimistic of persuading the former Charlton star, who has also been linked with Newcastle, to move to St Andrews.