In this paper we present the first results from an ongoing serendipitous survey aiming to identify X‐ray‐selected ‘normal’ galaxies [i.e. not dominated by active galactic nuclei (AGNs)] by combining archival XMM–Newton data with the Sloan Digital Sky Survey. In the first 4.5 deg2 of this program, we have identified a total of 11 ‘normal’ galaxy candidates (eight of them with optical spectroscopy) with fluxes fX(0.5–8 keV) ≈ 10−15–10−13 erg s−1 cm−2. These sources are selected to have low X‐ray‐to‐optical flux ratio (log fX/fopt≲−2), soft X‐ray spectral properties and optical spectra, when available, consistent with the presence of a stellar ionizing continuum. These sources comprise both early‐ and late‐type systems at redshifts z≲ 0.2 with luminosities LX(0.5–8 keV) ≈ 1039–1042 erg s−1. This data set provides the first tight constraint on the surface density of X‐ray‐selected ‘normal’ galaxies at relatively bright fluxes spanning 2 orders of magnitude (10−15–10−13 erg s−1 cm−2). The slope of the ‘normal’ galaxy log N–log S relation in the above flux range is estimated to be −1.4 ± 0.3, consistent with the Euclidean prediction. We also discuss the prospects of ‘normal’ galaxy studies at X‐ray wavelengths using both our continuously expanding survey and future X‐ray missions.