All the giant planets in the solar system host a large number of natural satellites. Moons in extrasolar systems are difficult to detect, but a Neptune-sized exomoon candidate has been recently found around a Jupiter-sized planet in the Kepler-1625bsystem. Due to their relative ease of detection, hot Jupiters (HJs), which reside in close orbits around their host stars with a period of a few days, may be very good candidates to search for exomoons. It is still unknown whether the HJ population can host (or may have hosted) exomoons. One suggested formation channel for HJs is high-eccentricity migration induced by a stellar binary companion combined with tidal dissipation. Here, we investigate under which circumstances an exomoon can prevent or allow high-eccentricity migration of a HJ, and in the latter case, if the exomoon can survive the migration process. We use both semianalytic arguments, as well as direct N-body simulations including tidal interactions. Our results show that massive exomoons are efficient at preventing high-eccentricity migration. If an exomoon does instead allow for planetary migration, it is unlikely that the HJ formed can host exomoons since the moon will either spiral onto the planet or escape from it during the migration process. A few escaped exomoons can become stable planets after the Jupiter has migrated, or by tidally migrating themselves. The majority of the exomoons end up being ejected from the system or colliding with the primary star and the host planet. Such collisions might nonetheless leave observable features, such as a debris disc around the primary star or exorings around the close-in giant.
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