Observations with the Cherenkov telescopes are in principle limited to clear sky conditions due to significant absorption of Cherenkov light by clouds. If the cloud level is high enough or the atmospheric transmission of the cloud is high, then high energy showers (with TeV energies) can still produce enough Cherenkov photons allowing detection by telescopes with large sizes and cameras with large field of view (FOV). In this paper, we study the possibility of observations of showers, induced by high-energy particles in the atmosphere, in the presence of clouds that are completely or partially opaque for Cherenkov radiation. We show how the image parameters of the Cherenkov light distribution on the telescope camera are influenced for different opacity and altitude of the cloud. By applying the Monte Carlo simulations, we calculate the scaled LENGTH and WIDTH parameters with the purpose to separate γ-ray and proton initiated showers in real data. We show, that the high level of the night sky background effects the selection efficiency of the γ-ray initiated showers. However, application of the higher image-cleaning level significantly improves expected quality factors. The estimated γ-ray selection efficiency for the detector with the camera field of view (FOV) limited to 8 is slightly better than for the camera with an unlimited FOV, although the number of identified γ-ray events is lower. We conclude that large Cherenkov telescopes with large FOV cameras can be used for observations of very high energy γ-rays in the presence of clouds. Consequently, the amount of useful data can be significantly enlarged.