The method presented here aims to assess the tsunami threat very rapidly after the occurrence of a large earthquake, using as input the parameters of the seismic source, and an approach based on Green's summation. We show that the main weakness of the approach (the need to consider only linear shallow water propagation) is largely compensated by the advantages in terms of computing performance and independence with respect to pre-computed scenarios. To test the approach and to illustrate its implementation in a real environment, we focus on the Sea of Oman, a tsunamigenic area characterized by Makran subduction zone which detailed structure is partially unknown and where secondary tsunami sources must also be taken into account, both for hazard studies and warning purposes. The potential source area is partitioned into a grid of unity water sources. A shallow water (SW) numerical model is used to pre-compute the corresponding empirical Green's functions on several points of interest located on the coasts of Iran, Pakistan and Oman. The comparison between Green's summation and the direct SW computation using the full resolution of the bathymetric grid shows that the accuracy is good enough for practical applications.