This paper studies an outsourcing problem where two service providers (suppliers) compete for the service contract from a client. The suppliers face uncertain cost for providing the service because they do not have perfect information about the client's type. The suppliers receive differential private signals about the client type and thus compete under asymmetric information. We first characterize the equilibrium of the supplier competition. Then we investigate two of the client's information sharing decisions. It is shown that less information asymmetry between the suppliers may dampen their competition. Therefore, the client does not necessarily have the incentive to reduce information asymmetry between the suppliers. We characterize the conditions under which leveling the informational ground is beneficial to the client. We also find that under the presence of information asymmetry (e.g., when the suppliers have different learning abilities), sharing more information with both suppliers may enhance the advantage of one supplier over the other and at the same time increase the upper bound of the suppliers' quotes in equilibrium. Consequently, the suppliers compete less aggressively and the client's payoff decreases in the amount of shared information. The findings from this study provide useful managerial implications on information management for outsourcing firms.