Aims. Outflows in quasars during the early epochs of galaxy evolution are an important part of the feedback mechanisms that potentially affect the evolution of the host galaxy. However, systematic millimetre (mm) observations of outflows are only now becoming possible with the advent of sensitive mm telescopes. In this study we used spectral stacking methods to search for a faint high-velocity outflow signal in a sample of [C II] detected, z ∼ 6 quasars. Methods. We searched for broad emission line signatures from high-velocity outflows for a sample of 26 z ∼ 6 quasars observed with the Atacama Large Millimeter Array (ALMA), with a detection of the [C II] line. The observed emission lines of the sources are dominated by the host galaxy, and outflow emission is not detected for the individual sources. We used a spectral line stacking analysis developed for interferometric data to search for outflow emission. We stacked both extracted spectra and the full spectral cubes. We also investigated the possibility that only a sub-set of our sample contributes to the stacked outflow emission. Results. We find only a tentative detection of a broad emission line component in the stacked spectra. When taking a region of about 2″ around the central position of the stacked cubes, the stacked line shows an excess emission due to a broad component of 1.1–1.5σ, but the significance drops to 0.4–0.7σ when stacking the extracted spectra from a smaller region. The broad component can be characterised by a line width of full width at half-maximum FWHM > 700 km s⁻¹. Furthermore, we find a sub-sample of 12 sources, the stack of which maximises the broad component emission. The stack of this sub-sample shows an excess emission due to a broad component of 1.2–2.5σ. The stacked line of these sources has a broad component of FWHM > 775 km s⁻¹. Conclusions. We find evidence suggesting the presence of outflows in a sub-sample of 12 out of 26 sources, which demonstrates the importance of spectral stacking techniques in tracing faint signal in galaxy samples. However, deeper ALMA observations are necessary to confirm the presence of a broad component in the individual spectra.