Design of infinite impulse response (IIR) digital filters to approximate some desired magnitude-frequency response is a classical research topic in signal processing. When a pole radius constraint is imposed, however, the problem becomes challenging and few solution methods are available. This paper converts the magnitude-response approximation problem into another problem that approximates the desired magnitude response and an accompanied phase response simultaneously. By iteratively updating the accompanied phase response, a solution to the original magnitude-response approximation problem can be obtained. A striking feature of the proposed method is that the pole radius constraint can be easily incorporated in the problem. Simulations and comparisons demonstrate the effectiveness of the method.