We present an extension of a linearized Coulomb collision operator, previously used in several Eulerian kinetic codes for like-species collisions and unlike-species collisions in the case where the backgrounds about which the linearization is made all are in collisional equilibrium, to the situation of interpenetrating plasma streams. In the latter case, the backgrounds cannot be taken to be in equilibrium and a significant generalization is required. Our development is targeted toward the Eulerian kinetic plasma code LOKI, which evolves the Vlasov-Poisson or Vlasov-Maxwell system in a Cartesian "2 + 2-dimensional" phase space. The extended operator has been implemented in a test code, and results of both quantitative verification and qualitative "realizability" tests are presented.