We present a kind of diffractive lens Zernike apodized photon sieves (ZAPS) whose structure is based on the combination of two concepts: apodized photon sieves and Zernike phase-contrast. Combined with the synchrotron light source, the ZAPS can be used as an objective for high-resolution phase-contrast X-ray microscopy in physical and life sciences. The ZAPS is a single optic unit that integrates the appropriate /2 radians phase shift through selective zone placement shifts in an apodized photon sieve. The focusing properties of the ZAPS can be easily controlled by apodizing its pupil function. An apodized photon sieve with Gaussian pupil is fabricated by lithographic technique and shows that the side-lobes are significantly suppressed at the expense of slightly widening the width of the main lobe.