A complete quantum-electrodynamical (QED) calculations of the interelectronic-interaction correction of first order in 1/Z were carried out to calculate the magnetic transition probabilities in high-Z He-like ions. The two-time Green-function method was used to derive the calculation formulas for the interelectronic-interaction corrections from the first principles of QED. The derived calculation formulas were then used for numerical evaluations of the magnetic transition probabilities in He-like ions. The numerical evaluation of these corrections to the magnetic transition probabilities was performed and the equivalence of the Feynman and Coulomb gauges were demonstrated.