Cooperative content delivery using multiple air interfaces (CCDMI) is a powerful solution to mitigate congestion in cellular networks. In CCDMI, the operator distributes content to selected users that further distribute it locally among its nearby users. However, a user that is capable of contributing to CCDMI might act selfishly and refuse to participate. Although the operator can encourage user participation by offering incentives, it has incomplete information about users' willingness to participate. In order to overcome this problem of adverse selection in CCDMI, we propose two contract-based methods under information asymmetry. In both methods, the operator designs a performance based contract set for the users that are capable of local content distribution. Using mathematical analysis, we show that the optimal contract under information asymmetry achieves close to optimal utility for the users and the operator, compared to the information symmetry case. Moreover, the users with high willingness to participate get positive utility and the users with low willingness get zero utility. Hence, by assigning contracts, the operator can motivate user participation, despite the information asymmetry between them. Our results verify that the proposed methods improve the system performance in terms of the utility of the operator and the users.