A two-dimensional quasilinear theory of saturation of the transverse magnetic Weibel instability in a collisionless plasma is developed. A closed system of equations is obtained to describe the evolution of the interacting spatial harmonics of the particle distribution function perturbations and magnetic field when the plasma anisotropy axis and wave vectors of the harmonics, as well as the electric field vectors in them, lie in the same calculation plane. Based on the numerical solution of this system, the dependence of the mean-square average of the saturating magnetic field on the initial anisotropies of the bi-Maxwellian and various bikappa velocity distributions of particles is found. It is shown that the magnitude of the saturating field substantially depends on the kappa parameter of the product bikappa particle distribution, i.e., on its energy profile, when the anisotropy parameters are small compared to unity. The results are compared with the published results of calculation by the less efficient particle-in-cell method.