Optical novae have recently been identified as the major class of supersoft X-ray sources in M 31 based on ROSAT and early XMM-Newton and Chandra observations. This paper reports on a search for X-ray counterparts of optical novae in M 31 based on archival Chandra HRC-I and ACIS-I as well as XMM-Newton observations of the galaxy center region obtained from July 2004 to February 2005. We systematically determine X-ray brightness or upper limit for counterparts of all known optical novae with outbursts between November 2003 to the end of the X-ray coverage. In addition, we determine the X-ray brightnesses for counterparts of four novae with earlier outbursts. For comparison with the X-ray data we created a catalogue of optical novae in M 31 based on our own nova search programs and on all novae reported in the literature. We collected all known properties and named the novae consistently following the CBAT scheme. We detect eleven out of 34 novae within a year after the optical outburst in X-rays. While for eleven novae we detect the end of the supersoft source phase, seven novae are still bright more than 1200, 1600, 1950, 2650, 3100, 3370 and 3380d after outburst. One nova is detected to turn on 50d, another 200d after outburst. Three novae unexpectedly showed short X-ray outbursts starting within 50d after the optical outburst and lasting only two to three months. The X-ray emission of several of the novae can be characterized as supersoft from hardness ratios and/or X-ray spectra or by comparing HRC-I count rates with ACIS-I count rates or upper limits. The number of detected optical novae at supersoft X-rays is much higher than previously estimated (>30%). We use the X-ray light curves to estimate the burned masses of the White Dwarf and of the ejecta.