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Aims. This study aims at constraining the origin of the nearby Type Ia supernovae (SNe), 2011fe and 2014J. The two most favoured scenarios for triggering the explosion of the white dwarf supernova progenitor is either mass loss from a non-degenerate companion or merger with another white dwarf. In the former, there could be a significant amount of leftover material from the companion at the centre of the supernova. Detecting such material would therefore favour the single-degenerate scenario. Methods. The left-over material from a possible non-degenerate companion can reveal itself after about one year, and in this study such material was searched for in the spectra of SN 2011fe (at 294 days after the explosion) using the Large Binocular Telescope and for SN 2014J using the Nordic Optical Telescope (315 days past explosion). The observations were interpreted using numerical models simulating the expected line emission from ablated material from the companion star. The spectral lines sought for are Hα, [O I] λ6300, and [Ca II] λλ7291,7324, and the expected width of these lines is ~1000 km s-1, which in the case of the [Ca II] lines blend to a broader feature. Results. No signs of Hα, [O I] λ6300, or [Ca II] λλ7291, 7324 could be traced for in any of the two supernovae. When systematic uncertainties are included, the limits on hydrogen-rich ablated gas are 0.003 M⊙ in SN 2011fe and 0.0085 M⊙ in SN 2014J, where the limit for SN 2014J is the second lowest ever, and the limit for SN 2011fe is a revision of a previous limit. Limits are also put on helium-rich ablated gas, and here limits from [O I] λ6300 provide the upper mass limits 0.002 M⊙ and 0.005 M⊙ for SNe 2011fe and 2014J, respectively. These numbers are used in conjunction with other data to argue that these supernovae can stem from double-degenerate systems or from single-degenerate systems with a spun-up/spun-down super-Chandrasekhar white dwarf. For SN 2011fe, other types of hydrogen-rich donors can very likely be ruled out, whereas a main-sequence donor system with large intrinsic separation is still possible for SN 2014J. Helium-rich donor systems cannot be ruled out for any of the two supernovae, but the expected short delay time for such progenitors makes this possibility less likely, especially for SN 2011fe. Published data for SNe 1998bu, 2000cx, 2001el, 2005am, and 2005cf are used to constrain their origin. We emphasise that the results of this study depend on the sought-after lines emerging unattenuated from the central regions of the nebula. Detailed radiative transfer calculations with longer line lists than are presently used are needed to confirm that this is, in fact, true. Finally, the broad lines of SNe 2011fe and 2014J are discussed, and it is found that the [Ni II] λ7378 emission is redshifted by ~+1300 kms-1, as opposed to the known blueshift of ~−1100 kms-1 for SN 2011fe. [Fe II] λ7155 is also redshifted in SN 2014J. SN 2014J belongs to a minority of SNe Ia that both have a nebular redshift of [Fe II] λ7155 and [Ni II] λ7378, and a slow decline of the Si II λ6355 absorption trough just after B-band maximum.