Context: The geometry and physical conditions in solar coronal loops are complicated and still not understood well. Recent high-resolution observations obtained with TRACE indicate the existence of sub-resolution transverse structuring not accessible to direct observation. This ingredient has not yet been taken into account in the previous theoretical models used for the study of transversal coronal loop oscillations and of their damping due to resonant conversion of energy. Aims: This study aims to assess the effect of the possibly unresolved internal structure of a coronal loop on the properties of its transverse oscillations and on the efficiency of resonant absorption as a damping mechanism of these oscillations. Methods: The equilibrium configuration of a single coronal loop with internal density structuring is modelled by considering the loop as composed of two very close, parallel, identical coronal slabs in Cartesian geometry. The period of the oscillation and the damping time are computed for the resonantly damped fundamental kink mode. These quantities are then compared to those obtained for two models for a single equivalent slab without internal density structuring. Results: We find that the period and the damping time of a coronal loop with internal density structuring change by less than 15%, when compared to the same oscillatory properties of a single coronal loop with either the same density contrast or a single coronal loop with the same total mass. Conclusions: Therefore the internal density structuring of a coronal loop does not affect its oscillatory properties very much. However, the sub-resolution structuring of a coronal loop with different densities in its components or with different widths could vary these results.