  We analyze a diffuse interface model that couples a viscous Cahn-Hilliard
equation for the phase variable with a diffusion-reaction equation for the
nutrient concentration. The system under consideration also takes into account
some important mechanisms like chemotaxis, active transport as well as nonlocal
interaction of Oono's type. When the spatial dimension is three, we prove the
existence and uniqueness of global weak solutions to the model with singular
potentials including the physically relevant logarithmic potential. Then we
obtain some regularity properties of the weak solutions when $t>0$. In
particular, with the aid of the viscous term, we prove the so-called
instantaneous separation property of the phase variable such that it stays away
from the pure states $\pm1$ as long as $t>0$. Furthermore, we study long-time
behavior of the system, by proving the existence of a global attractor and
characterizing its $\omega$-limit set.
