We cross-correlated the Chandra XASSIST and XMM-Newton Serendipitous Source Catalogues with the 2 degree Field Galaxy Redshift Survey ( 2dFGRS ) database . Our aim was to identify the most X-ray luminous ( L _ { X } > 10 ^ { 42 } erg s ^ { -1 } ) examples of galaxies in the local Universe whose X-ray emission is dominated by stellar processes rather than AGN activity ( ‘ normal ’ galaxies ) as well as to test the empirical criterion \log ( f _ { X } / f _ { O } ) < -2 for separating AGN from NGs . With XMM-Newton ( Chandra ) we covered an area of \sim 8.2 ( \sim 5.8 ) deg ^ { 2 } down to a flux limit of \sim 10 ^ { -15 } ( \sim 1.6 \times 10 ^ { -15 } ) erg cm ^ { -2 } s ^ { -1 } and found 18 ( 20 ) 2dFGRS galaxies . Using emission-line intensity ratios , we classified 6 2dFGRS spectra as star-forming , H ii nuclei , and 2 spectra as possible H ii nuclei . The rest of the objects are absorption-line galaxies and AGN , including 3 possible LINERs . No luminous ‘ normal ’ galaxies have been found but out of 19 ‘ normal ’ galaxies in this sample 5 H ii and 3 absorption-line galaxies have \log ( f _ { X } / f _ { O } ) > -2 . We performed a similar search in two nearby-galaxy samples from the literature . All 44 galaxies in the Zezas ( 27 ) sample have \log ( f _ { X } / f _ { O } ) < -2 and L _ { X } < 10 ^ { 42 } erg s ^ { -1 } . In the Fabbiano et al . ( 6 ) sample , out of a total of 170 ‘ normal ’ galaxies , we found 16 galaxies with \log ( f _ { X } / f _ { O } ) > -2 , the majority of which are massive ellipticals . Three of these have L _ { X } > 10 ^ { 42 } erg s ^ { -1 } .