  We present new catalogs of likely globular clusters (GCs) in 17 nearby spiral
galaxies studied as part of the PHANGS-HST Treasury Survey. The galaxies were
imaged in five broad-band filters from the near-ultraviolet through the $I$
band. PHANGS-HST has produced catalogs of stellar clusters of all ages by
selecting extended sources (from multiple concentration index measurements)
followed by morphological classification (centrally concentrated and symmetric
or asymmetric, multiple peaks, contaminant) by visually examining the V-band
image and separately by a machine-learning algorithm which classified larger
samples to reach fainter limits. From both cluster catalogs, we select an
initial list of candidate GCs to have $B-V \geq 0.5$ and $V-I \geq 0.73$~mag,
then remove likely contaminants (including reddened young clusters, background
galaxies misclassified by the neural network, and chance superpositions/blends
of stars) after a careful visual inspection. We find that $\approx86$ % of the
color-selected candidates classified as spherically symmetric, and $\approx68$
of those classified as centrally concentrated but asymmetric are likely to be
GCs. The luminosity functions of the GC candidates in 2 of our 17 galaxies, NGC
628 and NGC 3627, are atypical, and continue to rise at least 1~mag fainter
than the expected turnover near $M_V \sim -7.4$. These faint candidate GCs have
more extended spatial distributions than their bright counterparts, and may
reside in the disk rather than the bulge/halo, similar to faint GCs previously
discovered in M101. These faint clusters may be somewhat younger since the
age-metallicity degeneracy makes it difficult to determine precise cluster ages
from integrated colors once they reach $\approx1$~Gyr.
