Aims. We study the half-light radius versus black hole mass as well as the luminosity versus black hole mass relations in active galactic nuclei (AGN) when the disc is illuminated by the X-ray corona. Methods. We used KYNSED , a recently developed spectral model for studying broadband spectral energy distribution in AGN. We considered non-illuminated Novikov-Thorne discs and X-ray illuminated discs based on a Novikov-Thorne temperature radial profile. We also considered the case where the temperature profile is modified by a colour-correction factor. In the case of X-ray illumination, we assumed that the X-ray luminosity is equal to the accretion power that is dissipated to the disc below a transition radius and we computed the half-light radius and the disc luminosity for many black hole masses, as well as a wide range of accretion rates, black hole spins, X-ray luminosities and heights of the corona. Results. The half-light radius of X-ray illuminated radii can be up to ∼3.5 times greater than the radius of a standard disc, even for a non-spinning black hole, based on a wide range of model parameters – as long as the transition radius is larger than three times the radius of the innermost stable circular orbit and the coronal height is greater than ∼40  R g . This result is due to the fact that the absorbed X-rays act as a secondary source of energy, increasing the disc temperature, and mainly at large radii. Non-illuminated discs are consistent with observations, but only at the 2.5 σ level. On the other hand, X-ray illuminated discs can explain both the half-light radius-black hole mass as well as the luminosity-black hole mass relation in AGN, for a wide range of physical parameters. The range of the parameter space is broader in the case where we consider the colour-correction factor. X-ray illuminated discs can explain the data when we observe gravitationally lensed quasars mainly face-on, but also if the mean inclination angle is 60°. In addition, we show that the observed X-ray luminosity of the gravitationally lensed quasars is fully consistent with the X-ray luminosity that is necessary for heating the disc. Conclusions. X-ray disc illumination was proposed many years ago to explain various features that are commonly observed in the X-ray spectra of AGN. Recently, we showed that X-ray illumination of the accretion disc can also explain the observed UV/optical time-lags in AGN, while in this work, we show that the same model can also account for the quasar micro-lensing disc size problem. These results support the hypothesis of the disc X-ray illumination in AGN.