  We study nearly holomorphic Siegel Eisenstein series of general levels and
characters on $\mathbb{H}_{2n}$, the Siegel upper half space of degree $2n$. We
prove that the Fourier coefficients of these Eisenstein series (once suitably
normalized) lie in the ring of integers of $\mathbb{Q}_p$ for all sufficiently
large primes $p$. We also prove that the pullbacks of these Eisenstein series
to $\mathbb{H}_n \times \mathbb{H}_n$ are cuspidal under certain assumptions.
