Galaxies in compact groups tend to be deficient in neutral hydrogen compared to isolated galaxies of similar optical properties. In order to investigate the role played by a hot intragroup medium (IGM) for the removal and destruction of HI in these systems, we have performed a Chandra and XMM-Newton study of eight of the most HI deficient Hickson compact groups. Diffuse X-ray emission associated with an IGM is detected in four of the groups, suggesting that galaxy-IGM interactions are not the dominant mechanism driving cold gas out of the group members. No clear evidence is seen for any of the members being currently stripped of any hot gas, nor for galaxies to show enhanced nuclear X-ray activity in the X-ray bright or most HI deficient groups. Combining the inferred IGM distributions with analytical models of representative disc galaxies orbiting within each group, we estimate the HI mass loss due to ram pressure and viscous stripping. While these processes are generally insufficient to explain observed HI deficiencies, they could still be important for HI removal in the X-ray bright groups, potentially removing more than half of the ISM in the X-ray bright HCG 97. Ram pressure may also have facilitated strangulation through the removal of galactic coronal gas. In X-ray undetected groups, tidal interactions could be playing a prominent role, but it remains an open question whether they can fully account for the observed HI deficiencies.
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