We present new catalogs of likely globular clusters (GCs) in 17 nearby spiral galaxies studied as part of the PHANGS Hubble Space Telescope (HST) Treasury Survey. The galaxies were imaged in five broadband 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, and contaminants) by visually examining the V-band images 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 which have B − V ≥ 0.5 and V − I ≥ 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 ≈86% of the color-selected candidates classified as spherically symmetric and 68% of those classified as centrally concentrated but asymmetric are likely to be GCs. The luminosity functions of the GC candidates in two 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 ∼ −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 ≈1 Gyr.