  We extend the time-dependent Gutzwiller variational approach, recently
introduced by Schir\`o and Fabrizio, Phys. Rev. Lett. 105 076401 (2010), to
impurity problems. Furthermore, we derive a consistent theory for the steady
state, and show its equivalence with the previously introduced nonequilibrium
steady-state extension of the Gutzwiller approach. The method is shown to be
able to capture dissipation in the leads, so that a steady state is reached
after a sufficiently long relaxation time. The time-dependent method is applied
to the single orbital Anderson impurity model at half-filling, modeling a
quantum dot coupled to two leads. In these first exploratory calculations the
Gutzwiller projector is limited to act only on the impurity. The strengths and
the limitations of this approximation are assessed via comparison with state of
the art continuous time quantum Monte Carlo results. Finally, we discuss how
the method can be systematically improved by extending the region of action of
the Gutzwiller projector.
