The current flowing around a magnetic island and connected to its rotation with respect to the plasma is studied by means of a drift-kinetic approach. It is shown that the current due to a change in the precession frequency of the trapped particles in the island potential can compete with the standard polarization current for island rotation frequencies close to or below the diamagnetic frequency. The passing particles are found, on the contrary, to have little impact on the standard picture. The analytical results are shown to explain the features of the island current seen in numerical simulations.