The X-ray spectra of gamma-ray bursts (GRBs) can generally be described by an absorbed power law. The landmark discovery of thermal X-ray emission in addition to the power law in the unusual GRB 060218, followed by a similar discovery in GRB 100316D, showed that during the first thousand seconds after trigger the soft X-ray spectra can be complex. Both the origin and prevalence of such spectral components still evade understanding, particularly after the discovery of thermal X-ray emission in the classical GRB 090618. Possibly most importantly, these three objects are all associated with optical supernovae (SNe), begging the question of whether the thermal X-ray components could be a result of the GRB–SN connection, possibly in the shock breakout. We therefore performed a search for blackbody components in the early Swift X-ray spectra of 11 GRBs that have or may have associated optical SNe, accurately recovering the thermal components reported in the literature for GRBs 060218, 090618 and 100316D. We present the discovery of a cooling blackbody in GRB 101219B/SN2010ma, and in four further GRB–SNe we find an improvement in the fit with a blackbody which we deem possible blackbody candidates due to case-specific caveats. All the possible new blackbody components we report lie at the high end of the luminosity and radius distribution. GRB 101219B appears to bridge the gap between the low-luminosity and the classical GRB–SNe with thermal emission, and following the blackbody evolution we derive an expansion velocity for this source of the order of 0.4c. We discuss potential origins for the thermal X-ray emission in our sample, including a cocoon model which we find can accommodate the more extreme physical parameters implied by many of our model fits.