Recent observations with high spatial and temporal resolutions revealed different kinds of jets and flows in the solar atmosphere. Some of these jets can be twisted due to photospheric motions or eruption of new flux rope. Tangential velocity discontinuity near the boundaries of jets results in Kelvin-Helmholtz instability, which might contribute into plasma heating and turbulence. While the magnetic field component along the flow stabilizes the instability, a small transverse component may allow sub-Alfvénic motions to be unstable. We use magnetohydrodynamic equations in cylindrical geometry and derive the dispersion equations governing the dynamics of twisted and rotating magnetic flux tubes moving in untwisted and twisted external fields. Then, we solve the dispersion equations analytically and numerically and find thresholds for Kelvin-Helmholtz instability in both cases of the external field. Both analytical and numerical solutions show that the Kelvin-Helmholtz instability is suppressed in the twisted tube by the external axial magnetic field for sub-Alfvénic motions. However, even a small twist in the external magnetic field allows the Kelvin-Helmholtz instability to develop for any sub-Alfvénic motion. The unstable harmonics correspond to vortices with high azimuthal mode numbers that are carried by the flow. We estimate the instability criteria and growth times for different jets: spicules/macrospicules, x-ray/EUV jets, tornadoes etc.