Context. Finding the sources responsible for the hydrogen reionization is one of the most pressing issues in observational cosmology. Bright quasi-stellar objects (QSOs) are known to ionize their surrounding neighborhood, but they are too few to ensure the required HI ionizing background. A significant contribution by faint active galactic nuclei (AGNs), however, could solve the problem, as recently advocated on the basis of a relatively large space density of faint active nuclei at z > 4. Aims. This work is part of a long-term project aimed at measuring the Lyman Continuum escape fraction for a large sample of AGNs at z ~ 4 down to an absolute magnitude of M1450 ~ −23. We have carried out an exploratory spectroscopic program to measure the HI ionizing emission of 16 faint AGNs spanning a broad U − I color interval, with I ~ 21–23, and 3.6 < z < 4.2. These AGNs are three magnitudes fainter than the typical SDSS QSOs (M1450 ≲−26) which are known to ionize their surrounding IGM at z ≳ 4. Methods. We acquired deep spectra of these faint AGNs with spectrographs available at the VLT, LBT, and Magellan telescopes, that is, FORS2, MODS1-2, and LDSS3, respectively. The emission in the Lyman Continuum region, close to 900 Å rest frame, has been detected with a signal to noise ratio of ~10–120 for all 16 AGNs. The flux ratio between the 900 Å rest-frame region and 930 Å provides a robust estimate of the escape fraction of HI ionizing photons. Results. We have found that the Lyman Continuum escape fraction is between 44 and 100% for all the observed faint AGNs, with a mean value of 74% at 3.6 < z < 4.2 and − 25.1 ≲ M1450 ≲−23.3, in agreement with the value found in the literature for much brighter QSOs (M1450 ≲−26) at the same redshifts. The Lyman Continuum escape fraction of our faint AGNs does not show any dependence on the absolute luminosities or on the observed U − I colors of the objects. Assuming that the Lyman Continuum escape fraction remains close to ~75% down to M1450 ~ − 18, we find that the AGN population can provide between 16 and 73% (depending on the adopted luminosity function) of the whole ionizing UV background at z ~ 4, measured through the Lyman forest. This contribution increases to 25–100% if other determinations of the ionizing UV background are adopted from the recent literature. Conclusions. Extrapolating these results to z ~ 5–7, there are possible indications that bright QSOs and faint AGNs can provide a significant contribution to the reionization of the Universe, if their space density is high at M1450 ~ −23.