We find that the Fe III λλ2039-2113 spectral feature in quasars appears systematically redshifted by amounts accountable under the hypothesis of gravitational redshift induced by the central supermassive black hole (BH). Our analysis of 27 composite spectra from the BOSS survey indicates that the redshift and the broadening of the lines in the Fe III λλ2039-2113 blend roughly follow the expected correlation in the weak limit of Schwarzschild geometry for virialized kinematics. Assuming that the Fe III UV redshift provides a measure of \displaystyle \frac{{M}BH}}{R} ≤ft(\displaystyle \frac{{{Δ }}λ }{λ }≃ \displaystyle \frac{3}{2}\displaystyle \frac{G}{{c}2}\displaystyle \frac{{M}BH}}{R}\right) and using different estimates of the emitting region size, R (either from gravitational microlensing, reverberation mapping, or from the scaling of size with intrinsic quasar luminosity), we obtain masses for ten objects that are in agreement within uncertainties with previous mass estimates based on the virial theorem. Reverberation mapping estimates of the size of the Fe III λλ2039-2113 emitting region in a sample of objects would be needed to confirm the gravitational origin of the measured redshifts. Meanwhile, we present a tentative BH mass scaling relationship based on the Fe III λλ2039-2113 redshift useful to measure the BH mass of one individual object from a single spectrum.