  The late-stage evolution of the most massive stars such as {\eta} Carinae is
controlled by the effects of mass loss, which may be dominated by poorly
understood eruptive mass ejections. Understanding this population is
challenging because no true analogs of {\eta} Car have been clearly identified
in the Milky Way or other galaxies. We utilize Spitzer IRAC images of 7 nearby
(=<4 Mpc) galaxies to search for such analogs. We find 34 candidates with a
flat or rising mid-IR spectral energy distributions towards longer mid-infrared
wavelengths that emit 10^5 L_sun in the IRAC bands (3.6 to 8.0 micron) and are
not known to be background sources. Based on our estimates for the expected
number of background sources, we expect that follow-up observations will show
that most of these candidates are not dust enshrouded massive stars, with an
expectation of only 6+-6 surviving candidates. Since we would detect true
analogs of {\eta} Car for roughly 200 years post-eruption, this implies that
the rate of eruptions like {\eta} Car is less than the ccSN rate. It is
possible, however, that every M > 40 M_sun star undergoes such eruptions given
our initial results. In Paper II we will characterize the candidates through
further analysis and follow-up observations, and there is no barrier to
increasing the galaxy sample by an order of magnitude.
