Weak gravitational lensing is a powerful tool in modern cosmology. To accurately measure the weak lensing signal, one has to control the systematic bias on a small level. One of the most difficult problems is how to correct the smearing effect of the Point-Spread Function (PSF) on the shape of the galaxies. The chromaticity of PSF for a broad-band observation can lead to new subtle effects. Since the PSF is wavelength-dependent and the spectrum energy distributions between stars and galaxies are different, the effective PSF measured from the star images will be different from those that smear the galaxies. Such a bias is called color bias. We estimate it in the optical bands of the Chinese Space Station Survey Telescope from simulated PSFs, and show the dependence on the color and redshift of the galaxies. Moreover, due to the spatial variation of spectra over the galaxy image, another higher-order bias exists: color gradient bias. Our results show that both color bias and color gradient bias are generally below 0.1% in CSST. Only for small-size galaxies, one needs to be careful about the color gradient bias in the weak lensing analysis using CSST data.