We report periodic oscillations in the 15-year long optical light curve of the gravitationally lensed quasar Q J0158−4325 at z s = 1.29. The signal is enhanced during a high magnification microlensing event undergone by the fainter lensed image, B, of the quasar, between 2003 and 2010. We measure a period of P o = 172.6 ± 0.9 days, translating into 75.4 ± 0.4 days in the quasar frame. The oscillations have a maximum amplitude of 0.25 mag, and decrease concurrently with the smooth microlensing amplitude. We explore four scenarios to explain the origin of the periodicity: 1-the high magnification microlensing event is due to a binary star in the lensing galaxy, 2-Q J0158−4325 contains a massive binary black hole system in its final dynamical stage before merging, 3-the quasar accretion disk contains a bright inhomogeneity in Keplerian motion around the black hole, and 4-the accretion disk is in precession. Among these four scenarios, only a binary supermassive black hole can account for both the short observed period and the amplitude of the signal, through the oscillation of the accretion disk towards and away from high-magnification regions of a microlensing caustic. The short measured period implies that the semi-long axis of the orbit is ∼ 10 −3 pc, and the coalescence timescale is t coal ∼ 1000 years, assuming that the decay of the orbit is solely powered by the emission of gravitational waves. The probability of observing a system so close to coalescence, in a sample of only 30 monitored lensed quasars, suggests either a much larger population of supermassive black hole binaries than predicted, or, more likely, that some other mechanism significantly increases the coalescence timescale. Three tests of the binary black hole hypothesis include: i) the recurrence of oscillations in photometric monitoring during any future microlensing events in either image, ii) spectroscopic detection of Doppler shifts (up to ∼0.01c) associated to optical emission in the vicinity of the black holes, and iii) the detection of gravitational waves through Pulsar Timing Array experiments, such as the SKA, which will have the sensitivity to detect the ∼100 nano-hertz emission.