With the increasing prevalence of viral infections, the human T-cell leukemia virus (HTLV) is becoming a focal point of research. Of the four identified strains, HTLV-1 and HTLV-2 are particularly associated with various health issues. Both strains exhibit similar biological characteristics and transmission pathways, making them prevalent in specific high-risk populations, particularly among individuals who use injection drugs. HTLV-1 primarily targets the CD4$ ^{+} $ T cells, whereas HTLV-2 mainly affects the CD8$ ^{+} $ T cells. As far as we know, no mathematical model has been proposed to describe the within-host co-dynamics of HTLV-1 and HTLV-2. Therefore, this study presents a new mathematical framework to examine the within-host dynamics of HTLV-1 and HTLV-2 co-infection. Initially, the model's well-posedness is established by proving that the solutions remain both nonnegative and bounded over time. The equilibrium states and corresponding threshold conditions of the model are determined, and the criteria for the global asymptotic stability of each equilibrium are formulated. The global stability of the equilibria is analyzed using appropriate Lyapunov functions and LaSalle's invariance principle. These theoretical results are validated through numerical simulations. Additionally, sensitivity analysis of the basic reproduction numbers for HTLV-1 single infection ($ R_{1} $) and HTLV-2 single infection ($ R_{2} $) is performed to better understand the key parameters influencing co-infection dynamics. The study also explores the impact of CD8$ ^{+} $ T cell proliferation in the co-infection dynamics of HTLV-1 and HTLV-2, highlighting the importance of the CD8$ ^{+} $ T cell response in controlling the progression of HTLV-1. Furthermore, the impact of the viral infection rate on the co-infection dynamics of HTLV-1 and HTLV-2 is discussed. The results indicate that co-infection with HTLV-1 and HTLV-2 may increase the risk and severity of both viral infections.