Interatomic interactions and ordering in fcc Ni-rich Ni-Re alloys are studied by means of first-principles methods combined with statistical mechanics simulations based on the Ising Hamiltonian. First-principles calculations are employed to obtain effective chemical and strain-induced interactions, as well as ordering energies and enthalpies of formation of random and ordered Ni-Re alloys. Based on the nonmagnetic enthalpies of formation, we speculate that the type of ordering can be different in alloys with Re content less than 10 at.%. We demonstrate that effective chemical interactions in this system are quite sensitive to the alloy composition, atomic volume, and magnetic state. In statistical thermodynamic simulations, we have used renormalized interactions, which correctly reproduce ordering energies obtained in the direct total energy calculations. Monte Carlo simulations for Ni 0.91 Re 0.09 alloy show that there exists a strong ordering tendency of the (1 1 2 0) type leading to precipitation of the D1 a ordered structure at about 940 K. Our results for the atomic short-range order indicate, however, that the presently applied theory overestimates the strength of the ordering tendency compared to that observed in the experiment.