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The NASA mission OSIRIS-REx has been observing near-Earth asteroid (101955) Bennu in close proximity since December 2018. In this work, we investigate spectral phase reddening -- that is, the variation of spectral slope with phase angle -- on Bennu using spectra acquired by the OSIRIS-REx Visible and InfraRed Spectrometer (OVIRS) covering a phase angle range of 8-130$^{o}$. We investigate this process at the global scale and for some localized regions of interest (ROIs), including boulders, craters, and the designated sample collection sites of the OSIRIS-REx mission. Bennu has a globally negative spectra slope, which is typical of B-type asteroids. The spectral slope gently increases in a linear way up to a phase angle of 90$^{\circ}$, where it approaches zero. The spectral phase reddening is monotonic and wavelength-dependent with highest values in the visible range. Its coefficient is 0.00044 $μ$m$^{-1} ~deg^{-1}$ in the 0.55-2.5 $μ$m range. For observations of Bennu acquired at high phase angle (130$^{\circ}$), phase reddening increases exponentially. Similar behavior was reported in the literature for the carbonaceous chondrite Mukundpura in spectra acquired at extreme geometries. Some ROIs, including the sample collection site, Nightingale, have a steeper phase reddening coefficient than the global average, potentially indicating a surface covered by fine material with high micro-roughness. The gentle spectral phase reddening effect on Bennu is similar to that observed in ground-based measurements of other B-type asteroids, but much lower than that observed for other low-albedo bodies such as Ceres or comet 67P/Churyumov-Gerasimenko. Monotonic reddening may be associated with the presence of fine particles at micron scales and/or of particles with fractal structure that introduce micro- and sub-micro roughness across the surface of Bennu.