We show that black-hole High-Mass X-ray Binaries (HMXBs) with O- or B-type donor stars and relatively short orbital periods, of order one week to several months may survive spiral in, to then form Wolf-Rayet (WR) X-ray binaries with orbital periods of order a day to a few days; while in systems where the compact star is a neutron star, HMXBs with these orbital periods never survive spiral-in. We therefore predict that WR X-ray binaries can only harbor black holes. The reason why black-hole HMXBs with these orbital periods may survive spiral in is: the combination of a radiative envelope of the donor star, and a high mass of the compact star. In this case, when the donor begins to overflow its Roche lobe, the systems are able to spiral in slowly with stable Roche-lobe overflow, as is shown by the system SS433. In this case the transferred mass is ejected from the vicinity of the compact star (so-called isotropic re-emission mass loss mode, or SS433-like mass loss), leading to gradual spiral-in. If the mass ratio of donor and black hole is $>3.5$, these systems will go into CE evolution and are less likely to survive. If they survive, they produce WR X-ray binaries with orbital periods of a few hours to one day. Several of the well-known WR+O binaries in our Galaxy and the Magellanic Clouds, with orbital periods in the range between a week and several months, are expected to evolve into close WR-Black-Hole binaries,which may later produce close double black holes. The galactic formation rate of double black holes resulting from such systems is still uncertain, as it depends on several poorly known factors in this evolutionary picture. It might possibly be as high as $sim 10^{-5}$ per year.
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