We report the deepest radio interferometric observations of the closest Type Ia supernova in decades, SN 2014J, which exploded in the nearby galaxy M 82. These observations represent, together with radio observations of SNe 2011fe, the most sensitive radio studies of a Type Ia SN ever. We constrain the mass-loss rate from the progenitor system of SN 2014J to M ≲ 7.0 × 10-10 M⊙yr-1 (for a wind speed of 100 km s-1). Most single-degenerate scenarios, i.e., the wide family of progenitor systems where a red giant, main-sequence, or sub-giant star donates mass to a exploding white dwarf, are ruled out by our observations. On the contrary, our stringent upper limits to the radio emission from SN 2014 favor a double-degenerate scenario-involving two WD stars-for the progenitor system of SN 2014J, as such systems have less circumstellar gas than our upper limits. Thus, the evidence from SNe 2011fe and 2014J points in the direction of a double-degenerate scenario for both. Looking into the future, we note that the huge improvement in sensitivity of the SKA with respect to its predecessors will allow us to determine which progenitor scenario (single-degenerate vs. double-degenerate) applies to a large sample of nearby Type Ia SNe, thus unambiguously solving this issue.