Context. Fast radio bursts (FRBs) are millisecond radio transients observed at cosmological distances. The nature of their progenitors is still a matter of debate, although magnetars are invoked by most models. The proposed FRB–magnetar connection was strengthened by the discovery of an FRB-like event from the Galactic magnetar SGR J1935+2154. Aims. In this work we aim to investigate how prevalent magnetars such as SGR J1935+2154 are within FRB progenitors. Methods. To this end, we carried out an FRB search in a sample of seven nearby (< 12 Mpc) galaxies with the Northern Cross Radio Telescope for a total of 692 h. Results. We detected one 1.8 ms burst in the direction of M 101 with a fluence of 58 ± 5 Jy ms. Its dispersion measure of 303 pc cm⁻³ places it most likely beyond M 101. Considering that no significant detection comes indisputably from the selected galaxies, we place a 38 yr⁻¹ upper limit on the total burst rate (i.e. including the whole sample) at the 95% confidence level. This upper limit constrains the event rate per magnetar to λ_mag < 0.42 magnetar⁻¹ yr⁻¹ or, if combined with literature observations of a similar sample of nearby galaxies, it yields a joint constraint of λ_mag < 0.25 magnetar⁻¹ yr⁻¹. We also provide the first constraints on the expected rate of FRBs hypothetically originating from ultra-luminous X-ray (ULX) sources, since some of the galaxies observed during our observational campaign host confirmed ULXs. We obtain < 13 yr⁻¹ per ULX for the total sample of galaxies observed. Conclusions. Our results indicate that bursts with energies E > 10³⁴ erg from magnetars such as SGR J1935+2154 appear more rarely compared to previous observations and further disfavour them as unique progenitors for the cosmological FRB population. This provides support to the idea that there is a greater contribution from a population of more exotic magnetars not born via core-collapsed supernovae.