In this paper, we study the deadline-aware task data offloading in edge-cloud computing systems. The hard-deadline tasks strictly demand to be processed within their delay deadline, whereas the deadline can be relaxed for the soft-deadline tasks. Generally, edge computing aims to shorten the transmission delay between the remote cloud and the end-user, however, at the cost of limited computing capability. Therefore, it is challenging to decide where to offload the hard- and soft-deadline tasks based on the average delay and the service price set by the edge and cloud servers. Both edge and cloud servers aim to maximize their revenue by selling the computational resources at the optimal price. Interestingly, a Wardrop equilibrium is reached, considering that each task is considered independently to be offloaded to a suitable location. The numerical results demonstrate that the proposed price- and deadline-sensitive task offloading policy reaches the equilibrium and finds the optimal location for processing while maximizing the revenue of both edge and cloud servers.