We have imaged with HST's WFC3/UVIS the central 2.7$\times$2.7 arcmin$^2$ region of the giant elliptical galaxy M 87, using the ultraviolet filter F275W. In combination with archival ACS/WFC data taken through the F606W and F814W filters, covering the same field, we have constructed integrated-light UV-optical colors and magnitudes for 1460 objects, most of which are believed to be globular clusters belonging to M 87. The purpose was to ascertain whether the multiple-populations syndrome, ubiquitous among Galactic globular clusters (GCs), exists also among the M 87 family of clusters. To achieve this goal, we sought those GCs with exceptionally blue UV-to-optical colors, because helium-enriched sub-populations produce a horizontal-branch morphology that is well populated at high effective temperature. For comparison, integrated, synthetic UV$-$optical and purely optical colors and magnitudes have been constructed for 45 Galactic GCs, starting from individual-star photometry obtained with the same instruments and the same filters. We identify a small group of M 87 clusters exhibiting a radial UV$-$optical color gradient, representing our best candidate GCs hosting multiple populations with extreme helium content. We also find that the central spatial distribution of the bluer GCs is flattened in a direction parallel to the jet, while the distribution of redder GCs is more spherical. We release to the astronomical community our photometric catalog in F275W, F606W and F814W bands and the high-quality image stacks in the same bands. ; Comment: 20 pages (2 in Appendix), 15 figures (2 in low res), 2 tables, accepted for publication in ApJ