An achromatic phase-only superresolving pupil filter based on Bessel function is proposed and designed. The effect of the design parameters of the pupil filter on the superresolving figures of merit such as normalized compression ratio and Strehl ratio is discussed, respectively. Achromatic phase-only superresolving pupil filters with the materials of calcite crystal and optical quartz glass can realize achromatic superresolution in the ranges of 380~555 nm and 300~600 nm respectively with the normalized compression ratio of 0.83, and a relatively high Strehl ratio is also obtained.