Through offloading the computing tasks of the task nodes (TNs) to the fog nodes (FNs) located at the network edge, the fog network is expected to address the unacceptable processing delay and heavy link burden existed in current cloud-based networks. Unlike most existing researches based on the command-mode offloading and full capability report, this paper develops a general analytical model of the task scheduling among voluntary nodes (VNs) in fog networks, wherein the VNs voluntarily contribute their capabilities for serving their neighboring TNs. A novel delay-optimal task scheduling (DOTS) algorithm is proposed to obtain the delay-optimal offloading solution according to the reported capabilities of the VNs. Extensive simulations are carried out in a fog network, and the numerical results indicate that the proposed DOTS algorithm can effectively provide the optimal set of the helper nodes, subtask sizes, and the TN transmission power to minimize the overall task processing delay. Moreover, compared with the command-mode offloading, the voluntary-mode achieves more balanced offloading and a higher fairness level among the FNs.