  In perturbative consideration of the Yang--Mills gradient flow, it is useful
to introduce a gauge non-covariant term ("gauge-fixing term") to the flow
equation that gives rise to a Gaussian damping factor also for gauge degrees of
freedom. In the present paper, we consider a modified form of the gauge-fixing
term that manifestly preserves covariance under the background gauge
transformation. It is shown that our gauge-fixing term does not affect
gauge-invariant quantities as the conventional gauge-fixing term. The
formulation thus allows a background gauge covariant perturbative expansion of
the flow equation that provides, in particular, a very efficient computational
method of expansion coefficients in the small flow time expansion. The
formulation can be generalized to systems containing fermions.
