Since the onset of the HIV-1/AIDS epidemic in 1981, 75 million people have been infected with the virus, and the disease remains a public health crisis worldwide. Circular RNAs (circRNAs) are derived from excised exons and introns during backsplicing, a form of alternative splicing. The relevance of unconventional, non-capped, and non-poly(A) transcripts to transcriptomics studies remains to be routinely investigated. Knowledge gaps to be filled are the interface between host-encoded circRNAs and viral replication in chronically progressed patients and upon treatment with antiviral drugs. We implemented a bioinformatic pipeline and repurpose publicly archived RNA sequence reads from the blood of 19 HIV-1-positive patients that previously compared transcriptomes during viremia and viremia suppression by antiretroviral therapy (ART). The in silico analysis identified viremic patients’ circRNA that became undetectable after ART. The circRNAs originated from a subset of host genes enriched in the HDAC biological pathway. These circRNAs and parental mRNAs held in common a small collection of miRNA response elements (MREs), some of which were present in HIV-1 mRNAs. The function of the MRE-containing target mRNA enriched the RNA polymerase II GO pathway. To visualize the interplay between individual circRNA–miRNA–target mRNA, important for HIV-1 and potentially other diseases, an Interactive Circos tool was developed to efficiently parse the intricately competing endogenous network of circRNA–miRNA–mRNA interactions originating from seven circRNA singled out in viremic versus non-viremic patients. The combined downregulation of the identified circRNAs warrants investigation as a novel antiviral targeting strategy.