In this paper, we deal with the following system of nonlinear singular boundary value problems(BVPs) on time scale $mathbb{T}$ $${{{{{{x^{igtriangleupigtriangleup}(t)+f(t,;y(t))=0,;t{in}(a,;b)_{mathbb{T}},}atop{y^{igtriangleupigtriangleup}(t)+g(t,;x(t))=0,;t{in}(a,;b)_{mathbb{T}},}}atop{alpha_1x(a)-eta_1x^{igtriangleup}(a)=gamma_1x(sigma(b))+delta_1x^{igtriangleup}(sigma(b))=0,}}atop{alpha_2y(a)-eta_2y^{igtriangleup}(a)=gamma_2y(sigma(b))+delta_2y^{igtriangleup}(sigma(b))=0,}}$$ where $alpha_i$, $eta_i$, $gamma_i;{geq};0$ and $ ho_i=alpha_igamma_i(sigma(b)-a)+alpha_idelta_i+gamma_ieta_i$ > 0(i = 1, 2), f(t, y) may be singular at t = a, y = 0, and g(t, x) may be singular at t = a. The arguments are based upon a fixed-point theorem for mappings that are decreasing with respect to a cone. We also obtain the analogous existence results for the related nonlinear systems $x^{igtriangledownigtriangledown}(t)$ + f(t, y(t)) = 0, $y^{igtriangledownigtriangledown}(t)$ + g(t, x(t)) = 0, $x^{igtriangleupigtriangledown}(t)$ + f(t, y(t)) = 0, $y^{igtriangleupigtriangledown}(t)$ + g(t, x(t)) = 0, and $x^{igtriangledownigtriangleup}(t)$ + f(t, y(t)) = 0, $y^{igtriangledownigtriangleup}(t)$ + g(t, x(t)) = 0 satisfying similar boundary conditions.