For an integer m ≥ 3, a near m-gonal graph is a pair (Σ,E) consisting of a connected graph Σ and a set E of m-cycles of Σ such that each 2-arc of Σ is contained in exactly one member of E, where a 2-arc of Σ is an ordered triple (σ, τ, ε) of distinct vertices such that τ is adjacent to both σ and ε. The graph Σ is call (G, 2)-arc transitive, where G ≤ Aut(Σ), if G is transitive on the vertex set and on the set of 2-arcs of Σ. From a previous study it arises the question of when a (G, 2)-arc transitive graph is a near m-gonal graph with respect to a G-orbit on m-cycles. In this paper we answer this question by providing necessary and sufficient conditions in terms of the stabiliser of a 2-arc. Mathematics Subject Classification (2000). Primary 05C25; Secondary 20B25.