This paper explores the possibility of generation of a primordial magnetic field by a network of charged-current carrying cosmic strings. The field is created by vorticity, generated in the primordial plasma due to the strings’ motion and gravitational pull. In the case of superconducting strings formed at the breaking of grand unification, it is found that strong magnetic fields of high coherence can be generated in that way. Such fields could account for the observed galactic and intergalactic magnetic fields since they suffice to seed magnetic dynamos on galactic scales.