It is well known that the rotation of a magnetic island in the reference frame of plasma guiding centers generates parallel electron current outside the island, which is induced by the geodesic curvature of a magnetic field (Smolyakov et al 2007 Phys. Rev. Lett. 99 055002). It is shown in the present work that the surface part of this current located at the island separatrix can drive a pair of counter-propagating, tearing-parity, beta-induced Alfvén eigenmodes, which have the same helicity as that of the magnetic island and form a standing wave in the island frame. These Alfvénic modes can accompany tearing activity in tokamak discharges without energetic particles.