A linear stability study of cylindrical compressible gas jets in a moving incompressible viscous liquid medium subject to varicose disturbances is described. It was found that the gas jet is always unstable for a range of wavenumbers at any flow condition. When the gas and liquid velocity are not equal, temporal instability is enhanced by surface tension effects for small Weber numbers, and by aerodynamic interaction between the gas and liquid phase for high Weber numbers (where surface tension has a stabilizing influence). Increasing liquid viscosity always reduces the growth rate and the dominant wavenumber, whereas increasing gas density always increases gas jet instability. It was also found that the relative, rather than the absolute, velocity of the gas and liquid controls temporal instability. Increasing gas compressibility always increases the maximum growth rate and dominant wavenumber. On the other hand, for equal gas and liquid velocities, increasing surface tension always destabilizes, while increasing gas density always stabilizes, the gas jet. For absolute and spatial (or convective) instability, it was shown that the critical Weber number, separating the region of absolute from that of spatial instability, decreases monotonically as the liquid velocity is increased. For a stationary liquid medium, the gas jet is always absolutely unstable, and spatial instability does not exist, in contrast to liquid jets in a stationary gas medium. For sufficiently large liquid velocities, the gas jet is spatially unstable, whereas absolute instability disappears. Further, the absolute velocity of gas and liquid flow controls not only the growth of unstable disturbances, but also the characteristics of the instability. Increasing viscous effects tends to suppress absolute instability, while increasing both gas density and compressibility promotes absolute instability for small liquid velocities (however, their effect diminishes as liquid velocity is increased).