Self-consistent integral equations for the pair intramolecular and intermolecular correlation functions are derived from a general hierarchy of integral equations for chain-molecular liquids. These coupled equations are obtained by using superposition approximations for the triplet correlation functions, an approximate translational symmetry for the site-site intramolecular correlation functions and the equivalence of sites for intermolecular correlation functions. In addition to this self-consistent set of integral equations, the polymer reference interaction site model (PRISM) integral equation is also made self-consistent by coupling this intermolecular equation to the equations for the intramolecular correlation functions derived in the present theory. The intra- and intermolecular correlation functions of the self-consistent schemes considered in this work obey integral equations, and they are different from the other self-consistent schemes proposed in the literature. Self-consistent solutions for the structural properties, such as intra- and intermolecular correlation functions and structure factor, and macroscopic properties, such as chain expansion factor and thermodynamic functions of athermal polymer melts, are compared with available Monte Carlo results and other theories. For the properties examined, self-consistent solutions yield better results than the non-self-consistent calculations with ad hoc, ideal Gaussian inputs for the intramolecular correlation functions.