  In this paper we deal with diffusive relaxation limits of nonlinear systems
of Euler type modeling chemotactic movement of cells toward Keller--Segel type
systems. The approximating systems are either hyperbolic--parabolic or
hyperbolic--elliptic. They all feature a nonlinear pressure term arising from a
\emph{volume filling effect} which takes into account the fact that cells do
not interpenetrate. The main convergence result relies on compensated
compactness tools and is obtained for large initial data under suitable
assumptions on the approximating solutions. In order to justify such
assumptions, we also prove an existence result for initial data which are small
perturbation of a constant state. Such result is proven via classical
Friedrichs's symmetrization and linearization. In order to simplify the
coverage, we restrict to the two--dimensional case with periodical boundary
conditions.
