When an earthquake occurs and a tsunami threatens, rapid issuance of the first tsunami warning is important for timely evacuation of coastal residents. For tsunami early warning, estimates of an earthquake's hypocenter and magnitude are usually used. In Japan, the Japan Meteorological Agency (JMA) magnitude MJ, which is based on the observed displacement amplitude, is used to estimate the first tsunami warning. Slow tsunami earthquakes, such as the 1896 Meiji Sanriku earthquake, generate high tsunami waves but relatively small seismic waves. Thus, the use of MJ can cause underestimation of the size of such earthquakes and, therefore, lead to underestimation of the tsunami wave height. Quantitative understanding of the underestimation of slow tsunami earthquake magnitudes is needed, but local seismic records of slow tsunami earthquakes are scarce. In this study, we conducted spectrum analyses of teleseismic waves and used previously reported moment rate functions to construct synthetic local seismic wave records for slow tsunami earthquakes. First, we used data of earthquakes that occurred off the Japanese coast to confirm the validity of this method of constructing synthetic records. Then, we assumed tsunami earthquakes occurring off Miyagi Prefecture or the Sanriku Coast of Japan with the same moment rate functions as five major historical slow tsunami earthquakes, and compared our estimated magnitudes for these assumed earthquakes with the moment magnitudes (MW) of the five slow tsunami earthquakes. We found that MJ underestimated the size of the assumed earthquakes by 1 or more magnitude units when compared with MW. We also evaluated M100, a scale introduced after the 2011 Tohoku earthquake to supplement MJ and avoid underestimation of magnitude 9 class earthquakes. We found that M100 underestimated magnitudes by 0.5 or more magnitude units. Additionally, we suggest that amplitude distributions obtained from long-period seismic monitors, which were introduced to prevent underestimation of the magnitude of huge earthquakes, may be effectively used to estimate magnitudes of slow tsunami earthquakes.