  The gravitational self-force on a point particle moving in a vacuum
background spacetime can be expressed as an integral over the past worldline of
the particle, the so-called tail term. In this paper, we consider that piece of
the self-force obtained by integrating over a portion of the past worldline
that extends a proper time ${\Delta}{\tau}$ into the past, provided that
${\Delta}{\tau}$ does not extend beyond the normal neighborhood of the
particle. We express this ``quasi-local'' piece as a power series in the proper
time interval ${\Delta}{\tau}$. We argue from symmetries and dimensional
considerations that the $O({\Delta}{\tau}^0)$ and $O({\Delta}{\tau})$ terms in
this power series must vanish, and compute the first two non-vanishing terms
which occur at $O({\Delta}{\tau}^2)$ and $O({\Delta}{\tau}^3)$. The
coefficients in the expansion depend only on the particle's four velocity and
on the Weyl tensor and its derivatives at the particle's location. The result
may be useful as a foundation for a practical computational method for
gravitational self-forces in the Kerr spacetime, in which the portion of the
tail integral in the distant past is computed numerically from a mode sum
decomposition.
