In design of Variable Digital Filters (VDFs), Multi-Dimensional (M-D) polynomial approximation is well known as one of the powerful techniques. Although a number of methods for such VDF design can be seen in the literature, nothing has been proposed in this field with emphasis on high-accuracy filter structures with respect to the finite wordlength effects such as coefficient quantization error and roundoff noise. This paper presents M-D polynomial approximation-based VDFs with such a high-accuracy structure. To this end, we first decompose the pre-designed set of constant filters into second-order sections. Then, using the theory proposed in the literature, all of the second-order sections are described in terms of state-space representation of second-order Minimum Roundoff Noise (MRN) structure. Finally the resultant state-space coefficients are approximated by M-D polynomials based on spectral parameters. A numerical example demonstrates that the proposed method simultaneously attains the desired variable characteristics and approximately optimal performances with respect to the finite wordlength effects.