We find that the observed logN-logS relation of X-ray clusters (Ebeling et al. 1997; Rosati et al. 1997) can be reproduced remarkably well with a certain range of values for the fluctuation amplitude σ8 and the cosmological density parameter Ω0 in cold dark matter (CDM) universes (Kitayama & Suto 1997). The 1σ confidence limits on σ8 in the CDM models with n = 1 and h = 0.7 are expressed as (0.54 ± 0.02)Ω−0.35-0.82Ω0+0.55Ω2
0 (λ0 = 1 - Ω0) and (0.54 ± 0.02) Ω−0.28-0.91Ω0+0.68Ω2
0 (λ0 = 0), where n is the primordial spectral index, and h and λ0 are the dimensionless Hubble and cosmological constants. The errors quoted above indicate the statistical ones from the observed logN-logS only, and the systematic uncertainty from our theoretical modelling of X-ray flux in the best-fit value of σ8 is about 15%. In the case of n = 1, we find that the CDM models with (Ω0, λ0, h, σ8) ≃ (0.3, 0.7, 0.7, 1) and (0.45, 0, 0.7, 0.8) simultaneously account for the cluster logN-logS, X-ray temperature functions, and the normalization from the COBE 4 year data. The derived values assume the observations are without systematic errors, and we discuss in details other theoretical uncertainties which may change the limits on Ω0 and σ8 from the logN-logS relation. We have shown the power of this new approach which will become a strong tool as the observations attain more precision.