We report results from a deep search for redshifted HI 21 cm absorption from eight damped Lyman a absorbers (DLAs) detected in our earlier optical survey for DLAs towards radio-loud quasars. HI 21 cm absorption was detected from the z = 2.192 DLA towards TXS 2039+187, only the sixth case of such a detection at z > 2, while upper limits on the HI 21 cm optical depth were obtained in sixother DLAs at z > 2. Our detection of HI 21 cm absorption in the eighth system, at z=2.289 towards TXS 0311+430, has been reported earlier. We also present high spatial resolution images of the background quasars at frequencies close to the redshifted HI 21 cm line frequency, allowing us to estimate the covering factor of each DLA, and then determine its spin temperature Ts. For three non-detections of HI 21 cm absorption, we obtain strong lower limits on the spin temperature, Ts ≳ 790 K, similar to the bulk of the high-z DLA population; three other DLAs yield weak lower limits, Ts > 140Kanekar N., Braun R., Roy N., 2011, ApJ, 737, L33-400 K. However, for the two DLAs with detections of HI 21 cm absorption, the derived spin temperatures are both low, Ts = (160 ± 35) × (f/0.35) K for the z = 2.192 DLA towards TXS 2039+187 and Ts = (72 ± 18) × (f/0.52) K for the z = 2.289 DLA towards TXS 0311+430. These are the first two DLAs at z > 1 with Ts values comparable tothose obtained in local spiral galaxies. Basedon the observed correlations between spin temperature and metallicity and velocity spread and metallicity in DLAs, we suggest that the hosts of the two absorbers are likely to be massive, high-metallicity galaxies. © 2012 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.