The Einstein Telescope (ET) is a third-generation gravitational wave (GW) observatory that will be constructed in the coming decade. It consists of an underground, triangular laser interferometer and is designed to observe GWs - ripples in the space-time continuum - due to e.g. binary black hole or binary neutron star mergers. In order to observe GWs caused by mergers at larger distance or with a larger mass, the sensitivity and operating frequency range (above 10 Hz) of current second-generation GW observatories is insufficient. The design sensitivity of the ET is about one order of magnitude lower and the target frequency is as low as 3 Hz. As many anthropogenic noise sources interfere with its operation at low frequencies, ET will be constructed underground in a seismically quiet region. A dominant noise source at low frequencies is seismic Newtonian noise (NN): waves propagating in the soil generate density fluctuations which result in gravitational attraction and corresponding motion of the test masses inside the interferometer. Since one cannot shield these gravitational forces, the only remedy is to estimate seismic NN from the measured seismic field through wavefield reconstruction and subtract it from the interferometer data. Seismic NN can be computed by inserting the density fluctuations due to seismic displacements in Newton's second law of motion and integrating over the soil domain around the test mass. This study presents a numerical model in which the soil domain surrounding the test mass is discretized with finite elements. Gaussian quadrature is used to numerically evaluate the integral expressions. The model is validated by predicting NN due to plane P- and S-waves propagating in a homogeneous medium, for which analytical expressions are available. The influence of the size and shape of the cavern surrounding the test mass on the NN is also studied. It is further demonstrated how NN due to anthropogenic sources such as road and railway traffic can be predicted for the ET candidate site Euregio Meuse-Rhine.