It has been shown that the irradiation of a gas target with nanoclusters by a short (tens of femtoseconds), relativistically intense, circularly polarized laser pulse results in the appearance of magnetic dipoles, which are oriented in the direction of the propagation of the laser pulse and produce a quasi-static regular magnetic field with an amplitude to tenths of the field of the initial laser pulse. At a sufficient concentration of clusters, after the end of the laser pulse, the magnetic moments of the dipoles undergo vibrations and the magnetic field oscillates, which is accompanied by the appearance of a short (few-cycle) secondary terahertz radiation. The condition for the appearance of such a short terahertz pulse is that the cyclotron frequency of an electron on average over the volume of the magnetic field is higher than the inverse lifetime of the cluster.