Sinc-Nyström methods for Fredholm integral equations of the second kind have been independently proposed by Muhammad et al. and Rashidinia-Zarebnia. They also gave error analyses, but the results did not claim the convergence of their schemes in a precise sense. This is because in their error estimates there remained an unestimated term: the norm of the inverse of the coefficient matrix of the resulting linear system. In this paper, we estimate the term theoretically to complete the convergence estimate of their methods. Furthermore, we also prove the boundedness of the condition number of each coefficient matrix.