Black hole (BH) spins in low-mass X-ray binaries cover a range of values that can be explained by accretion after BH birth. In contrast, the three BH spin measurements in high-mass X-ray binaries (HMXBs) show only values near the maximum and likely have a different origin connected to the BH stellar progenitor. We explore here two possible scenarios to explain the high spins of BHs in HMXBs: formation in binaries that undergo mass transfer (MT) during the main sequence (MS; Case-A MT), and very close binaries undergoing chemically homogeneous evolution (CHE). We find that both scenarios are able to produce high-spin BHs if internal angular momentum (AM) transport in the progenitor star after its MS evolution is not too strong (i.e., weak coupling between the stellar core and its envelope). If instead efficient AM transport is assumed, we find that the resulting BH spins are always too low with respect to observations. The Case-A MT model provides a good fit for the BH spins, the masses of the two components, and the final orbital periods for two of the three BHs in HMXBs with measured spins. For one of them, the mass predicted for the BH companion is significantly lower than observed, but this depends strongly on the assumed efficiency of MT. The CHE models predict orbital periods that are too large for all three cases considered here. We expect the Case-A MT to be much more frequent at the metallicities involved, so we conclude that the Case-A MT scenario is preferred. Finally, we predict that the stellar companions of HMXBs formed through the Case-A MT have enhanced nitrogen surface abundances, which can be tested by future observations.