We present a study of the old globular clusters (GC) using archival F606W and F814W HST/ACS images of 19 Magellanic-type dwarf Irregular (dIrr) galaxies found in nearby (2 - 8 Mpc) associations of only dwarf galaxies. All dIrrs have absolute magnitudes fainter than or equal to the SMC (Mv = -16.2 mag). We detect 50 GC candidates in 13 dIrrs, of which 37 have (V-I) colors consistent with blue (old, metal-poor) GCs (bGC). The luminosity function (LF) of the bGCs in our sample peaks at Mv = -7.41 +/- 0.22 mag, consistent with other galaxy types. The width of the LF is sigma = 1.79 +/- 0.31 which is typical for dIrrs, but broader than the typical width in massive galaxies. The half-light radii and ellipticities of the GCs in our sample (rh ~ 3.3 pc, e ~ 0.1) are similar to those of old GCs in the Magellanic Clouds and to those of Old Halo (OH) GCs in our Galaxy, but not as extended and spherical as the Galactic Young Halo (YH) GCs (rh ~ 7.7 pc, e ~ 0.06). The e distribution shows a turnover rather than a power law as observed for the Galactic GCs. This might suggest that GCs in dIrrs are kinematically young and not fully relaxed yet. The present-day specific frequencies (SN) span a broad range: 0.3 < SN < 11. Assuming a dissipationless age fading of the galaxy light, the SN values would increase by a factor of ~ 2.5 to 16, comparable with values for early-type dwarfs (dE/dSphs). A bright central GC candidate, similar to nuclear clusters of dEs, is observed in one of our dIrrs: NGC 1959. This nuclear GC has luminosity, color, and structural parameters similar to that of wCen and M54, suggesting that the latter might have their origin in the central regions of similar Galactic building blocks. A comparison between properties of bGCs and Galactic YH GCs, suspected to have originated from similar dIrrs, is performed.
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