Tsunami warning centres inform the population who are located near and around coastal areas by determining the size and location of an earthquake. The Pacific Tsunami Early Warning System has used P-wave moment magnitude (Mwp) for the warning system for many years. A moment magnitude determination from the P-wave arrivals has the advantage of estimating the size of an earthquake within a couple of minutes, compared to other estimation procedures. The Eastern Mediterranean region is tectonically active area and has experienced many tsunami earthquakes in history. In this paper, the moment magnitude determination technique was used to calculate the moment magnitudes of events that occurred in the Eastern Mediterranean and near the coasts of Turkey with magnitudes between 5.5 and 6.8 and epicentral distance between about 100 and 1700 km. Mwp was calculated using seismic stations that were installed in the area and the number of stations used changed event by event. Their numbers were in the range of 11 to 71. Calculated Mwp values showed good compatibility with Global Centroid Moment Tensor Mw. The results of this study are encouraging for the early warning system that will be set up in the near future in the region.