  Black hole (BH) spins in low-mass X-ray binaries (LMXBs) 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 mass transfer. 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.
