The 3-D inverse modelling of a magnetic anomaly measured over the NW submarine edifice of the volcanic island of Gran Canaria revealed a large, reversely-magnetized, elongated structure following an ENE-WSW direction, which we interpreted as a sill-like magmatic intrusion emplaced during the submarine growth of this volcanic island, with a volume that could represent up to about 20% of the whole island. The elongated shape of this body suggests the existence of a major crustal fracture in the central part of the Canary Archipelago which would have favoured the rapid ascent and emplacement of magmas during a time span from 0.5 to 1.9 My during a reverse polarity chron of the Earth’s magnetic field prior to 16 Ma. The agreement of our results with those of previous gravimetric, seismological and geodynamical studies strongly supports the idea that the genesis of the Canary Islands was conditioned by a strike-slip tectonic framework probably related to Atlas tectonic features in Africa. These results do not contradict the hotspot theory for the origin of the Canary magmatism, but they do introduce the essential role of regional crustal tectonics to explain where and how those magmas both reached the surface and built the volcanic edifices.