Quantum mechanical tunneling effects are investigated using an extension of the full multiple spawning (FMS) method. The FMS method uses a multiconfigurational frozen Gaussian ansatz for the wave function and it allows for dynamical expansion of the basis set during the simulation. Basis set growth is controlled by allowing this expansion only when the dynamics signals impending failure of classical mechanics, e.g., nonadiabatic and/or tunneling effects. Previous applications of the FMS method have emphasized the modeling of nonadiabatic effects. Here, a new computational algorithm that accounts for tunneling effects is introduced and tested against exact solution of the Schrödinger equation for two multi-dimensional model problems. The algorithm first identifies the tunneling events and then determines the initial conditions for the newly spawned basis functions. Quantitative agreement in expectation values, tunneling doublets and tunneling splitting is demonstrated for a wide range of conditions.