This paper addresses the problem of reducing the delivery time of data messages to cellular users using instantly decodable network coding (IDNC) with physical-layer rate awareness. While most of the existing literature on IDNC does not consider any physical layer complications and abstracts the model as equally slotted time for all users, this paper proposes a cross-layer scheme that incorporates the different channel rates of the various users in the decision process of both the transmitted message combinations and the rates with which they are transmitted. The consideration of asymmetric rates for receivers reflects more practical application scenarios and introduces a new tradeoff between the choice of coding combinations for various receivers and the broadcasting rates. The completion time minimization problem in such a scenario is first shown to be intractable. The problem is thus approximated by reducing, at each transmission, the increase of an anticipated version of the completion time. This paper solves the problem by formulating it as a maximum weight clique problem over a newly designed rate-aware IDNC graph. The highest weight clique in the created graph being potentially not unique, this paper further suggests a multi-layer version of the proposed solution to improve the obtained results from the employed completion time approximation. Simulation results indicate that the cross-layer design largely outperforms the uncoded transmissions strategies and the classical IDNC scheme.