  A finite group $G$ is called Cayley integral if all undirected Cayley graphs
over $G$ are integral, i.e., all eigenvalues of the graphs are integers. The
Cayley integral groups have been determined by Kloster and Sander in the
abelian case, and by Abdollahi and Jazaeri, and independently by Ahmady, Bell
and Mohar in the non-abelian case. In this paper we generalize this class of
groups by introducing the class $\mathcal{G}_k$ of finite groups $G$ for which
all graphs $\mathrm{Cay}(G,S)$ are integral if $|S| \le k$. It will be proved
that $\mathcal{G}_k$ consists of the Cayley integral groups if $k \ge 6;$ and
the classes $\mathcal{G}_4$ and $\mathcal{G}_5$ are equal, and consist of:\ (1)
the Cayley integral groups, (2) the generalized dicyclic groups
$\mathrm{Dic}(E_{3^n} \times \mathbb{Z}_6),$ where $n \ge 1$.
