  We define the quantum-incoherent relative entropy of coherence
($\mathcal{QI}$ REC) of quantum channels in the framework of the resource
theory by using the Choi-Jamiolkowsky isomorphism. Coherence-breaking channels
are introduced as free operations and their corresponding Choi states as free
states. We also show the relationship between the coherence of channel and the
quantum discord and find that basis-dependent quantum asymmetric discord can
never be more than the $\mathcal{QI}$ REC for any quantum channels. {Also}, we
prove the $\mathcal{QI}$ REC is decreasing for any divisible quantum incoherent
channel and we also claim it can be considered as the quantumness of quantum
channels. Moreover, we demonstrate that for qubit channels, the relative
entropy of coherence (REC) can be equivalent to the REC of their corresponding
Choi states and the basis-dependent quantum symmetric discord can never exceed
the coherence.
