Circular synthetic aperture radar (CSAR), which can observe the region of interest for a long time and from multiple angles, offers the opportunity for moving-target detection (MTD). However, traditional MTD methods cannot effectively solve the problem of high probability of false alarm (PFA) caused by strong clutter. To mitigate this, a novel, three-step scheme combining clutter background extraction, multichannel clutter suppression, and the degree of linear consistency of radial velocity interferometric phase (DLRVP) test is proposed. In the first step, the spatial similarity of the scatterers and the correlation between sub-aperture images are fused to extract the strong clutter mask prior to clutter suppression. In the second step, using the data remaining after elimination of the background clutter in Step 1, an amplitude-based detector with higher processing gain is utilized to detect potential moving targets. In the third step, a novel test model based on DLRVP is proposed to further reduce the PFA caused by isolated strong scatterers. After the above processing, almost all false alarms are excluded. Measured data verified that the PFA of the proposed method is only 20% that of the comparison method, with improved detection of slow and weakly moving targets and with better robustness.