The nearly 33 yr long-term radio light curve obtained with the Metsahovi Radio Observatory 14 m telescope at 37 GHz and the recent 12.7 yr gamma-ray light curve of the blazar S5 0716+714 at 0.1-300 GeV from the Fermi Large Area Telescope (Fermi-LAT) were analyzed by using the Lomb-Scargle periodogram and the weighted wavelet Z-transform techniques. In the radio light curve, we discovered a possible quasi-periodic oscillation (QPO) signal of about 352 +/- 23 days at a confidence level of similar to 3 sigma. We recalculated the periodicity and its significance in a chosen time range that has higher variability and denser sampling, and then found that the significance had increased to a confidence level of 99.996% (∼ 4.1σ). This QPO component was further confirmed by fitting a linear autoregressive integrated moving average model to the selected radio light curve. A possible QPO of 960 +/- 80 days at a 99.35% level (∼ 2.7σ) was found in the gamma-ray light curve, which generally agrees with the earlier QPO claims of S5 0716+714. This paper discusses possible mechanisms for this potential year-like QPO. One possibility is a pure geometrical scenario with blobs moving helically inside the jet. Another is a supermassive binary black hole involving a gravitational wave-driven regime. In the latter scenario, we derived a milliparsec separation in the binary system that undergoes coalescence within a century due to the emission of low-frequency gravitational waves.