This work is concerned with a system of two singular viscoelastic equations with general source terms and nonlocal boundary conditions. We discuss the stabilization of this system under a very general assumption on the behavior of the relaxation function k_{i}, namely, ki′(t)≤−ξi(t)Ψi(ki(t)),i=1,2.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} k_{i}^{\\prime }(t)\\le -\\xi _{i}(t) \\Psi _{i} \\bigl(k_{i}(t)\\bigr),\\quad i=1,2. \\end{aligned}$$ \\end{document} We establish a new general decay result that improves most of the existing results in the literature related to this system. Our result allows for a wider class of relaxation functions, from which we can recover the exponential and polynomial rates when k_{i}(s) = s^{p} and p covers the full admissible range [1, 2).