We propose a method for near-field tsunami forecasting from data acquired by cabled offshore ocean bottom tsunami meters (OBTMs) in real time. We first invert tsunami waveforms recorded at OBTMs to estimate the spatial distribution of initial sea-surface displacements in the tsunami source region without making any assumptions about fault geometry and earthquake magnitude. Then, we synthesize the coastal tsunami waveforms from the estimated sea-surface displacement distribution. To improve the reliability of the tsunami forecasting, we use updated OBTM data to repeat the forecast calculation at 1-min intervals. We tested our method by simulating the 1896 Sanriku tsunami earthquake, which caused a devastating tsunami with maximum runup height of 38 m along the Pacific coast of northeastern Japan. Instead of real OBTM records, proxies were used. The simulation demonstrated that our method provided accurate estimations of coastal arrival times and amplitudes of the first peak of the tsunami more than 20 min before the maximum amplitude wave reached the coastal site nearest to the source. We also applied the method to real data of a small tsunami that was caused by a local earthquake and successfully forecasted the tsunami at coastal tide stations. We found that accuracy of our estimated coastal tsunami amplitudes can be affected by the spatial relationship between the tsunami source and the offshore observation stations. Our numerical simulation showed that even more accurate tsunami amplitude forecasts would be achieved by deployment of additional offshore stations separated by a distance comparable to the trench-parallel length of the tsunami source.