Let G be a finite simple graph, and JG denote the binomial edge ideal of G. In this paper, we first compute the v-number of binomial edge ideals corresponding to Cohen-Macaulay closed graphs. As a consequence, we obtain the v-number for paths. For cycle and binary tree graphs, we obtain a sharp upper bound for v(JG) using the number of vertices of the graph. We characterize all connected graphs G with v(JG) = 2. We show that for a given pair (k,m),k ≤ m, there exists a graph G with an associated monomial edge ideal I having v-number equal to k and regularity m. We also show that if 2k ≤ m, then there exists a binomial edge ideal with v-number k and regularity m. Finally, we compute v-number of powers of binomial edge ideals with linear resolution, thus proving a conjecture on the v-number of powers of a graded ideal having linear powers, for the class of binomial edge ideals.