  Anti-solar differential rotation profiles have been found for decades in
numerical simulations of convective envelopes of solar-type stars. These
profiles are characterized by a slow equator and fast poles (i.e., reversed
with respect to the Sun) and have been found in simulations for high Rossby
numbers (slow rotators). Rotation profiles like this have been reported
observationally in evolved stars, but have never been unambiguously observed
for cool solar-type stars on the main sequence. In this context, detecting this
regime in main-sequence solar-type stars would improve our understanding of
their magnetorotational evolution. The goal of this study is to identify the
most promising cool main-sequence stellar candidates for anti-solar
differential rotation in the \textit{Kepler} sample. First, we introduce a new
theoretical formula to estimate fluid Rossby numbers, $Ro_{\rm f}$, of
main-sequence solar-type stars, from observational quantities, and taking the
influences of the internal structure and metallicity into account. We obtain a
list of the most promising stars that are likely to show anti-solar
differential rotation. We identify two samples: one at solar metallicity,
including 14 targets, and another for other metallicities, including 8 targets.
We find that the targets with the highest $Ro_{\rm f}$ are likely to be early-G
or late-F stars at about log$_{10}g=4.37$~dex. We conclude that cool
main-sequence stellar candidates for anti-solar differential rotation exist in
the \textit{Kepler} sample. The most promising candidate is KIC~10907436, and
two other particularly interesting candidates are the solar analog KIC~7189915
and the seismic target KIC~12117868. Future characterization of these 22 stars
is expected to help us understand how dynamics can impact magnetic and
rotational evolution of old solar-type stars at high Rossby number.
