Since January 2021, the diffusion of the most propagated SARS-CoV-2 variants in France (UK variant 20I/501Y.V1 (lineage B.1.1.7), 20H/H501Y.V2 (lineage B.1.351) and 20J/H501Y.V3 (lineage P.1)) were urgently screened, needing a surveillance with an RT-PCR screening assay. In this study, we evaluated one RT-PCR kit for this screening (ID SARS-CoV-2/UK/SA Variant Triplex®, ID Solutions, Grabels, France) on 2207 nasopharyngeal samples that were positive for SARS-CoV-2. Using ID Solutions kit, 4.1% (92/2207) of samples were suspected to belonged to B.1.351 or P.1 variants. Next-generation sequencing that was performed on 67.4% (62/92) of these samples confirmed the presence of a B.1.351 variant in only 75.8% of the samples (47/62). Thirteen samples belonged to the UK variant (B.1.1.7), and two to A.27 with N501Y mutation. The thirteen with the UK variant presented one mutation in the S-gene, near the ΔH69/ΔV70 deletion (S71F or A67S), which impacted the detection of ΔH69/ΔV70 deletion. Using another screening kit (PKampVariantDetect SARS-CoV-2 RT-PCR combination 1 and 3® PerkinElmer, Waltham, MA, USA) on the misidentified samples, we observed that the two mutations, S71F or A67S, did not impact the detection of the UK variant. In conclusion, this study highlights the limitations of the screening strategy based on the detection of few mutations/deletions as well as it not being able to follow the virus evolution.