The appropriate mode pairs of primary and double-frequency circumferential guided waves (CGWs) have been investigated and selected for generation of the cumulative second harmonics, which are applicable for quantitative assessment of damage/degradation in a circular tube. The selection criteria follow the requirements: the higher efficiency of cumulative second-harmonic generation (SHG) of primary CGW propagation, and the larger response sensitivity of cumulative SHG to material damage/degradation [characterized by variation in the third-order elastic (TOE) constants]. The acoustic nonlinearity parameter β of CGW propagation and the change rate of normalized β versus the TOE constants of tube material are, respectively, used to describe the efficiency of SHG and its response sensitivity to damage/degradation. Based on the selection criteria proposed, all the possible mode pairs of primary and double-frequency CGWs satisfying the phase velocity matching have been numerically examined. It has been found that there are indeed some mode pairs of CGW propagation with the larger values both of β and the change rate of normalized β versus the TOE constants. The CGW mode pairs found in this paper are of practical significance for quantitative assessment of damage/degradation in the circular tube.