Context. One of the novelties of Gaia data release 3 (DR3) with respect to the previous data releases is the publication of the multiband light curves for about 1 million active galactic nuclei (AGN) and of the values of some parameters characterising their variability properties. Aims. The goal of this work is the creation of a catalogue of variable AGN, the selection of which is based on Gaia data only. Methods. We first present the implementation of the methods used to estimate the variability parameters within the framework of a specific object study module for AGN (SOS-AGN). We then describe the selection procedure that led to the definition of the high-purity Gaia variable AGN sample and analyse the properties of the selected sources. We started from a sample of millions of sources, which were identified as AGN candidates using 11 different classifiers based on variability processing. Because the focus is on the variability properties, we first defined some pre-requisites in terms of number of data points in the G band and mandatory variability parameters. A series of filters was then applied using only Gaia data and the Gaia Celestial Reference Frame 3 (Gaia-CRF3) sample as a reference. Results. The resulting Gaia AGN variable sample, named GLEAN, contains about 872'000 objects, more than 21'000 of which are new identifications. We checked for the presence of contaminants by cross-matching the selected sources with a variety of galaxies and stellar catalogues. The completeness of GLEAN with respect to the variable AGN in the last Sloan Digital Sky Survey (SDSS) quasar catalogue is a 47%, while that based on the variable AGN of the Gaia-CRF3 sample is a 51%. The set of filters applied to the sources selected by SOS-AGN to increase the sample purity reduced the source number by about 37%. From both a comparison with other AGN catalogues and an investigation of possible contaminants, we conclude that purity can be expected to be above 95%. The multi-wavelength properties of these sources are investigated. In particular, we estimate that a 4% of them are radio-loud. We finally explore the possibility to evaluate the time-lags between the flux variations of the multiple images of strongly lensed quasars, and show one case.