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Context. The full third Gaia data release will provide, for the first time, the calibrated spectra obtained with the blue and red Gaia slitless spectrophotometers (BP and RP, respectively). Gaia is a very complex mission and cannot be considered as a single instrument, but rather as many instruments. The two lines of sight with wide fields of view introduce strong variations of the observations across the large focal plane with more than one hundred different detectors. The main challenge when facing Gaia spectral calibration is that no lamp spectra or flat fields are available during the mission. Also, the significant size of the line spread function with respect to the dispersion of the prisms produces alien photons contaminating neighbouring positions of the spectra. This makes the calibration special and different from standard approaches. Aims. This work gives a detailed description of the internal calibration model for the spectrophotometric data used to obtain the content of the Gaia catalogue. The main purpose of the internal calibration is to bring all the epoch spectra onto a common flux and pixel (pseudo-wavelength) scale, taking into account variations over the focal plane and with time, producing a mean spectrum from all the observations of the same source. Methods. In order to describe all observations on a common mean flux and pseudo-wavelength scale, we constructed a suitable representation of the internally calibrated mean spectra via basis functions, and we described the transformation between non-calibrated epoch spectra and calibrated mean spectra via a discrete convolution, parametrising the convolution kernel to recover the relevant coefficients. Results. The model proposed here for the internal calibration of the Gaia spectrophotometric observations is able to combine all observations into a mean instrument to allow the comparison of different sources and observations obtained with different instrumental conditions along the mission and the generation of mean spectra from a number of observations of the same source. We derived a calibration model that can handle the self-calibrating nature of the problem. The output of this model provides the internal mean spectra, not as a sampled function (flux and wavelength), but as a linear combination of basis functions, although sampled spectra can easily be derived from them.