We define a measure of the quantumness of correlations, based on the operative task of local broadcasting of a bipartite state. Such a task is feasible for a state if and only if it corresponds to a joint classical probability distribution, or, in other terms, it is strictly classically correlated. A gap, defined in terms of quantum mutual information, can quantify the degree of failure in fulfilling such a task, therefore providing a measure of how non-classical a given state is. We are led to consider the asymptotic average mutual information of a state, defined as the minimal per-copy mutual information between parties, when they share an infinite amount of broadcast copies of the state. We analyze the properties of such quantity, and find that it satifies many of the properties required for an entanglement measure. We show that it lies between the quantum- and the classical-conditioned versions of squashed entanglement. The non-vanishing of asymptotic average mutual information for entangled states may be interpreted as a signature of monogamy of entanglement.