Star clusters are valuable indicators of galaxy evolution, offering insights into the buildup of stellar populations across cosmic time. Understanding the intrinsic star cluster populations of dwarf galaxies is particularly important given these systems' role in the hierarchical growth of larger systems. We use data from Euclid's Early Release Observation programme to study star clusters in two star-forming dwarf irregular galaxies in the Local Group, NGC 6822 and IC 10 [M⋆ ~ (1─4) × 108 M⊙]. With Euclid, star clusters are resolved into individual stars across the main bodies and haloes of both galaxies. Through visual inspection of the IE images, we uncover 30 new star cluster candidates in NGC 6822 and 16 in IC 10, ranging from compact to diffuse extended clusters. We compile and re-evaluate previously identified literature candidates, resulting in final combined catalogues of 52 (NGC 6822) and 71 (IC 10) cluster candidates with confidence-based classifications. We present homogeneous photometry in IE, YE, JE, and HE, and in archival UBVRI data, alongside size measurements and properties derived from the spectral energy distribution fitting code BAGPIPES. Through synthetic cluster injection, we conclude our sample is ~50% complete to M ≲ 103 M⊙ for ages ≲100 Myr, and to M ≲ 2 × 104 M⊙ for ages of ~10 Gyr. We find that IC 10 has more young clusters than NGC 6822, and its young clusters extend to higher masses, consistent with its starburst nature. We find several old massive (≳105 M⊙) clusters in both dwarfs, including an exceptional cluster in NGC 6822's outskirts with a mass of 1.3 × 106 M⊙, nearly twice as massive as any other old cluster in either galaxy. In NGC 6822, we also identify a previously undetected, old, and extended cluster (Rh = 12.4 ± 0.11 pc). Using well-defined criteria, we identify 11 candidate GCs in NGC 6822 and nine in IC 10. Both galaxies have high specific frequencies (SN) for their luminosities but remain consistent with the known GC scaling relationships in the low-luminosity regime.