We forecast the expected population of active galactic nuclei (AGN) observable in the Euclid Wide Survey (EWS) and Euclid Deep Survey (EDS). Starting from an X-ray luminosity function (XLF), we generated volume-limited samples of the AGN expected in the Euclid survey footprints. Each AGN was assigned a spectral energy distribution (SED) appropriate for its X-ray luminosity and redshift, with perturbations sampled from empirical distributions. The photometric detectability of each AGN was assessed via mock observations of the assigned SED. We estimate 40 million AGN will be detectable in at least one Euclid band in the EWS and 0.24 million in the EDS, corresponding to surface densities of 2.8 × 103 deg-2 and 4.7 × 103 deg-2. The relative uncertainty on our expectation for Euclid detectable AGN is 6.7% for the EWS and 12.5% for the EDS, driven by the uncertainty of the XLF. Employing Euclid-only colour selection criteria on our simulated data we select a sample of 4.8 × 106 (331 deg-2 ) AGN in the EWS and 1.7 × 104 (346 deg-2 ) in the EDS, amounting to 10% and 8% of the AGN detectable in the EWS and EDS. Including ancillary Rubin/LSST bands improves the completeness and purity of AGN selection. These data roughly double the total number of selected AGN to comprise 21% and 15% of the Euclid detectable AGN in the EWS and EDS. The total expected sample of colour-selected AGN contains 6.0 × 106 (74%) unobscured AGN and 2.1 × 106 (26%) obscured AGN, covering 0.02 ≤ z ≲ 5.2 and 43 ≤ log10(Lbol/erg s-1 ) ≤ 47. With these simple colour cuts expected surface densities are already comparable to the yield of modern X-ray and mid-infrared surveys of similar area. The EWS sample is most comparable to the WISE C75 AGN selection and the EDS sample is most similar to the yield of the collated Spitzer cryogenic surveys when considering Euclid bands alone, or the XXL-3XLSS survey AGN sample when also considering selection with ancillary optical bands. We project that 15% (7.6%) of the total Euclid detectable population in the EWS (EDS) will exhibit X-ray fluxes that could be detected in the XMM-COSMOS survey, showing that the vast majority of Euclid-detected AGN would not be detectable in modern medium-depth X-ray surveys.