  We establish relations between Gorenstein projective precovers linked by
Frobenius functors. This is motivated by an open problem that how to find
general classes of rings for which modules have Gorenstein projective
precovers. It is shown that if $F:\C\rightarrow\D$ is a separable Frobenius
functor between abelian categories with enough projective objects, then every
object in $\C$ has a Gorenstein projective precover provided that every object
in $\D$ has a Gorenstein projective precover. This result is applied to
separable Frobenius extensions and excellent extensions.
