The progenitors of hydrogen-poor core-collapse supernovae (SNe) of Types Ib, Ic and IIb are believed to have shed their outer hydrogen envelopes either by extremely strong stellar winds, characteristic of classical Wolf-Rayet stars, or by binary interaction with a close companion star. The exact nature of the progenitors and the relative importance of these processes are still open questions. One relatively unexplored method to constrain the progenitors is to search for high-mass X-ray binaries (HMXBs) at SN locations in pre-explosion X-ray observations. In an HMXB, one star has already exploded as a core-collapse SN, producing a neutron star or a stellar mass black hole. It is likely that the second star in the system will also explode as an SN, which should cause a detectable long-term change in the system's X-ray luminosity. In particular, a pre-explosion detection of an HMXB coincident with an SN could be informative about the progenitor's nature. In this paper, we analyse pre-explosion ACIS observations of 18 nearby Type Ib, Ic and IIb SNe from the Chandra X-ray observatory public archive. Two sources that could potentially be associated with the SN are identified in the sample. Additionally we make similar post-explosion measurements for 46 SNe. Although our modelling indicates that progenitor systems with compact binary companions are probably quite rare, studies of this type can in the future provide more stringent constraints as the number of discovered nearby SNe and suitable pre-explosion X-ray data are both increasing.