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In order to considering the integrality of nearly holomorphic (vector-valued)Siegel modular forms, we introduce nearly Siegel modular forms and study theirintegrality. We show that the integrality of nearly Siegel modular forms interms of their Fourier expansion implies the integrality of their CM values.Furthermore, we show that there exists a one-to-one correspondence betweenintegral nearly Siegel modular forms and integral nearly holomorphic ones. Bythese results, the integrality of CM values holds for nearly holomorphic Siegelmodular forms and for nearly overconvergent p-adic Siegel modular forms.

Integrality of nearly (holomorphic) Siegel modular forms