The v-number of a graded ideal I ⊆ R, denoted by v(I), is the minimum degree of a polynomial f for which I : f is a prime ideal. Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021) studied the v-number of edge ideals. In this paper, we study the v number of the cover ideal J(G) of a graph G. The main result shows that v(J(G)) ≤ reg(R/J(G)) for any simple graph G, which is quite surprising because, for the case of edge ideals, this inequality does not hold. Our main result relates v(J(G)) with the Cohen-Macaulay property of R/I(G), where I(G) denotes the edge ideal of G. We provide an infinite class of connected graphs, which satisfy v(J(G)) = reg(R/J(G)). Also, we show that for every positive integer k, there exists a connected graph Gk such that reg(R/J(Gk)) − v(J(Gk)) = k. Also, we explicitly compute the v-number of cover ideals of cycles.