We report deep EVN and eMERLIN observations of the Type Ia SN 2014J in the nearby galaxy M82. Our observations represent, together with JVLA observations of SNe 2011fe and 2014J, the most sensitive radio studies of Type Ia SNe ever. By combining data and a proper modeling of the radio emission, we constrain the mass-loss rate from the progenitor system of SN 2014J to $\dot{M} \lesssim 7.0\times 10^{-10} {\,{M_{\odot } \,\rm yr^{-1}}}$ (for a wind speed of 100 km s-1). If the medium around the supernova is uniform, then n ISM lesssim 1.3 cm-3, which is the most stringent limit for the (uniform) density around a Type Ia SN. Our deep upper limits favor a double-degenerate (DD) scenario involving two WD stars for the progenitor system of SN 2014J, as such systems have less circumstellar gas than our upper limits. By contrast, most single-degenerate (SD) scenarios, i.e., the wide family of progenitor systems where a red giant, main-sequence, or sub-giant star donates mass to an exploding WD, are ruled out by our observations. (While completing our work, we noticed that a paper by Margutti et al. was submitted to The Astrophysical Journal. From a non-detection of X-ray emission from SN 2014J, the authors obtain limits of $\dot{M} \lesssim 1.2 \times 10^{-9}$ M ☉ yr-1 (for a wind speed of 100 km s-1) and n ISM lesssim 3.5 cm-3, for the ρ∝r -2 wind and constant density cases, respectively. As these limits are less constraining than ours, the findings by Margutti et al. do not alter our conclusions. The X-ray results are, however, important to rule out free-free and synchrotron self-absorption as a reason for the radio non-detections.) Our estimates on the limits on the gas density surrounding SN2011fe, using the flux density limits from Chomiuk et al., agree well with their results. Although we discuss the possibilities of an SD scenario passing observational tests, as well as uncertainties in the modeling of the radio emission, the evidence from SNe 2011fe and 2014J points in the direction of a DD scenario for both.