We use the thread model for linear chains of interacting monomers and the "polymer reference interaction site model" formalism to determine the monomer-monomer pair correlation function h(mm)(r) for dilute and semidilute polymer solutions, over a range of temperatures from very high (where the chains behave as self-avoiding walks) to below the theta temperature, where phase separation sets in. An inversion procedure, based on the HNC integral equation, is used to extract the effective pair potential between "average" monomers on different chains. An accurate relation between h(mm)(r), h(cc)(r) (the pair correlation function between the polymer centers of mass (c.m.)), and the intramolecular form factors is then used to determine hcc(r) and subsequently extract the effective c.m.-c.m. pair potential v(cc)(r) by a similar inversion procedure. vcc(r) depends on temperature and polymer concentration, and the predicted variations are in reasonable agreement with recent simulation data, except at very high temperatures and below the theta temperature.