In this study, we investigate two mathematical models, formulated using delay differential equations, to capture HIV-1 transmission dynamics. Both models incorporate CD4 ${}^{+}$T cells and macrophages as target cells and saturated non-linear infection transmission terms. The first model introduces a discrete time delay, while the second employs a distributed delay to reflect more realistic biological insights. Our theoretical analysis explores key properties such as the basic reproduction number, non-negativity, boundedness, and the existence of equilibria. The global stability analysis of the disease-free and endemic equilibrium are analysed using the LaSalle invariance principle by proposing Lyapunov functions. We conducted sensitivity analysis to find the significant parameters related to the infection dynamics. Numerical simulations are performed to validate our theoretical results and visualize the behavior of the system under different parametric conditions. This study shows the critical role of time delays in shaping HIV-1 infection dynamics. Incorporation of such delays into mathematical models are essential for accurately capturing the progression of the disease and proposing effective intervention strategies for HIV-1.