We investigate connected normal 2-geodesic transitive Cayley graphs $\mathrm{Cay}(T,S)$. We first prove that if $\mathrm{Cay}(T,S)$ is neither cyclic nor $K_{4\vert 2\vert}$, then $\langle a\rangle\backslash\{1\}\subsetneq S$ for all $a \in S$. Next, as an application, we give a reduction theorem proving that each graph in this family which is neither a complete multipartite graph nor a bipartite 2-arc transitive graph, has a normal quotient that is either a complete graph or a Cayley graph in the family for a characteristically simple group. Finally we classify complete multipartite graphs in the family.

05C25

Cayley graph, normal 2-geodesic transitivity