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The spreading of HIV or HTLV-I among the cells has received the great attention in recent modelling study to explore the virus infection dynamics. The co-infection of HIV and HTLV-I with the effect of Cytotoxic T-lymphocytes (CTLs) immune response is also important from epidemiological point of view. To identify the co-infection scenario of HIV and HTLV-I with the CTLs effect we proposed in this paper a six compartmental ODE-model with uninfected, HIV-infected, HTLV-I infected CD4+T cells and free HIV virus particles with HIV specific CTLs and HTLV-I specific CTLs. The rates of infection of the cases are considered here saturated type and proliferation rate of uninfected and HIV infected CD4+ T-cells are of logistic terms. To establish the well-posedness of the model we have shown that the solution of the proposed model is non-negative and bounded. We obtain the basic reproduction number which is the maximum of the HIV-related reproduction and the HTLV-I related reproduction number. Along with the disease free equilibrium point the system contains other seven endemic equilibrium points containing infection by single disease or both. Analytically, we establish the local and global stability conditions of the equilibrium points and also we establish that the system experiences transcritical bifurcation by the generation of only HIV or HTLV-I infected endemic equilibrium point. Using numerical simulations, we validate the theoretical results and found two infection paths, one initiating with HIV and other with HTLV-I, both cases ultimately become co-infected. Finally, using the optimal control analysis we found the optimal policy for treatment using AVR, RTI & PI for HIV or AZT for HTLV-I control and lastly concluded by some recommendations. • A co-infected HIV & HTLV-I infection model in vivo. • Proliferation of CD4+ T-cells are taken in logistic form. • Incidence rate is taken as saturated type. • Verified competitive exclusive principle. • Spreading path has been studied using Flow-diagram. • Three types of control strategies are investigated. [ABSTRACT FROM AUTHOR]