We test whether the masses of old globular clusters (GCs) in dwarf galaxies are consistent with the same initial mass spectrum as young massive clusters (YMCs) in nearby star-forming galaxies. The most massive GCs of dwarf galaxies are compared to their expected masses when drawing from the Schechter-type initial cluster mass function (ICMF) of YMCs. It is found that the most massive GCs of galaxies in the stellar mass range M★, gal= 107–109 M⊙ are consistent with the same initial mass spectrum as YMCs in about 90 per cent of the cases, suggesting that their formation mechanisms were the same. For the remaining 10 per cent, the most massive clusters are nuclear GCs, which have been able to grow to higher masses through further merging after their initial formation ended. Because the effects of cluster disruption are weaker for more massive clusters, we estimate that up to one-third of the metal-poor GCs in the Milky Way may have a nuclear origin, while the remaining two-thirds formed through the same process as YMCs in the local Universe. A lognormal ICMF is inconsistent with observed GCs at a 99.6 per cent confidence level.