There is ample evidence of twisted magnetic structures in the solar corona. This motivates us to consider the magnetic twist as the cause of Alfv´en frequency continuum in coronal loops, which can support the resonant absorption as a rapid damping mechanism for the observed coronal kink magnetohydrodynamic (MHD) oscillations. We model a coronal loop with a straight cylindrical magnetic flux tube, which has constant but different densities in the interior and exterior regions. The magnetic field is assumed to be constant and aligned with the cylinder axis everywhere except for a thin layer near the boundary of the flux tube, which has an additional small magnetic field twist. Then, we investigate a number of possible instabilities that may arise in our model. In the thin tube thin boundary approximation, we derive the dispersion relation and solve it analytically to obtain the frequencies and damping rates of the fundamental (l = 1) and first/second overtone (l = 2, 3) kink (m = 1) MHD modes. We conclude that the resonant absorption by the magnetic twist can justify the rapid damping of kink MHD waves observed in coronal loops. Furthermore, the magnetic twist in the inhomogeneous layer can cause deviations from P1/P2 = 2 and P1/P3 = 3, which are comparable with the observations.
