  In this paper we show that an expanding or steady gradient Ricci soliton
warped product $B^n\times_f F^m$, $m>1$, whose warping function $f$ reaches
both maximum and minimum must be a Riemannian product. Moreover, we present a
necessary and sufficient condition for constructing a gradient Ricci soliton
warped product. As an application, we present a class of expanding Ricci
soliton warped product having as a fiber an Einstein manifold with non-positive
scalar curvature. We also discuss some obstructions to this construction,
especially in the case when the base of the warped product is compact.
