Using integral field spectroscopy (IFS) we have observed QSOs which were previously known to have intervening DLAs, to search for emission lines in the field surrounding the QSOs. The sample consists of 13 DLAs plus 7 sub-DLAs at $z>2$, from which we find emission line candidates at the expected wavelengths for 8 systems, implying a detection frequency of 40%. Emission lines detected at the DLA redshifts are interpreted as being due to Ly$\alpha$ lines from galaxies associated with the DLA clouds. Candidates are detected at $3-4\sigma$ levels in both narrow-band images and spectra created from the data cubes. We find no preference for candidate detection associated with either high or low metallicity DLAs. In comparison with the properties of previously known DLA galaxies, the candidates show similar distributions of line fluxes and line widths, but their impact parameters are slightly different, suggesting H I cross sections larger than 10 kpc.