An n-by-n real symmetric matrix H=[h(i,j)] is said to be a Hankel matrix if h(i,j)=h(i-1,j+1), for each i=2,...,n and j=1,...,n-1. The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a lists lambda={lambda(1), ... ,lambda(n)} of real numbers is the spectrum of an n-by-n symmetric nonnegative matrix H. In this paper, we search for conditions on the list lambda={lambda 1, ....,lambda(n)} for the matrix H to be Hankel. For n = 3, sufficient conditions are established. In particular, a necessary and sufficient condition is obtained if lambda is a list of three nonnegative numbers. Also, if sigma(n )(i=1)lambda(i)=0, we give conditions for realizability by a Hankel matrix. Finally, we present a special type of list that can serve as the spectrum of a Hankel nonnegative matrix with positive trace. Several of our results are constructive and provide a Hankel realizing matrix.