The paper presents a paraxial theory of the relativistic cross-focusing of two coaxial Gaussian laser beams of different frequencies in a homogeneous plasma. We discuss the self-focusing of a weaker laser beam in the plasma due to the optical inhomogeneities introduced by another stronger copropagating laser beam. In the presence of the second stronger beam (Pcr21<P2<Pcr22), the plasma behaves as an oscillatory waveguide for the first, weaker, beam (P1<Pcr11) as it propagates in the plasma. When both the beams are strong (Pcr11,21<P1,2<Pcr12,22), the nonlinearities introduced by the relativistic effect are additive in nature, such that one beam can undergo oscillatory self-focusing and the other simultaneously defocusing, and vice versa. A comparison reveals that cross-focusing due to relativistic nonlinearity is possible for a wider range of powers of the laser pulses than is cross-focusing due to ponderomotive nonlinearity. Relativistic cross-focusing is important in plasma beat-wave excitation and collective laser particle accelerators.