We have analyzed the light curves of 527 short gamma-ray bursts (GRBs) registered by the SPI-ACS, SPI and IBIS/ISGRI experiments of INTEGRAL observatory totally to search for precursors. Both the light curves of each 527 individual burst and the averaged light curve of 372 brightest SPI-ACS bursts have been analyzed. In a few cases, we have found and investigated precursor candidates using SPI-ACS, SPI and IBIS/ISGRI of INTEGRAL, GBM and LAT of Fermi data. No convincing evidence for the existence of precursors of short GRBs has been found. A statistical analysis of the averaged light curve for the sample of brightest short bursts has revealed no regular precursor. Upper limits for the relative intensity of precursors have been estimated. We show that the fraction of short GRBs with precursors is less than 0.4% of all short bursts, detected by INTEGRAL.