We have conducted a near-infrared monitoring campaign at the UK Infrared Telescope (UKIRT), of the Local Group spiral galaxy M33 (Triangulum). The main aim was to identify stars in the very final stage of their evolution, and for which the luminosity is more directly related to the birth mass than the more numerous less-evolved giant stars that continue to increase in luminosity. The most extensive data set was obtained in the K band with the UIST instrument for the central 4 × 4 arcmin2 (1 kpc2) – this contains the nuclear star cluster and inner disc. These data, taken during the period 2003–2007, were complemented by J- and H-band images. Photometry was obtained for 18 398 stars in this region; of these, 812 stars were found to be variable, most of which are asymptotic giant branch (AGB) stars. Our data were matched to optical catalogues of variable stars and carbon stars and to mid-infrared photometry from the Spitzer Space Telescope. In this first of a series of papers, we present the methodology of the variability survey and the photometric catalogue – which is made publicly available at the Centre de Données astronomiques de Strasbourg – and discuss the properties of the variable stars. The most dusty AGB stars had not been previously identified in optical variability surveys, and our survey is also more complete for these types of stars than the Spitzer survey.