We theoretically investigate a new mechanism of interfacial current in a two-layer system consisting of a magnetic insulator and an adjacent non-magnetic metal. The mechanism is based on Rashba spin-orbit interaction in the metal layer near the interface where the magnetic insulator induces non-zero magnetization. The rotation of the magnetization of the magnetic insulator induces the alternating interfacial current in the non-magnetic metal. Coordinate and time dependencies of the current and induced magnetization in the non-magnetic layer are calculated using quasi-energy approach assuming ballistic conductivity in non-magnetic metal. It is found that the current displays sizable magnitude within the metal layer over a distance which significantly exceeds a region with non-zero spin-orbit interaction. Both the current and the spin density induced in the metal layer demonstrate oscillatory dependence as a function of the distance from the interface due to the interference of the incoming and reflected waves.