A wall-attached cube immersed in a zero pressure gradient boundary layer is studied by means of Direct Numerical Simulations (DNS) at various Reynolds numbers ReH (based on the cube height and the free-stream velocity) ranging from 500 to 3000. The cube is either immersed in a laminar boundary layer (LBL) or in a turbulent boundary layer (TBL), with the aim to understand the mechanisms of the unsteady flow structures generated downstream of the wall-attached cube. The mean locations of the stagnation and recirculation points around the cube immersed in a TBL are in good agreement with reference experimental and numerical data, even if in those studies the cube was immersed in a turbulent channel. In the TBL simulation, a vortex shedding can be identified in the energy spectra downstream of the cube, with Strouhal number of St=0.14. However, the frequency of the vortex shedding is different in the LBL simulations, showing a significant dependence on the Reynolds number. Furthermore, in the TBL simulation, a low frequency peak with St=0.05 can be observed far away from the boundary layer, at long streamwise distances from the cube. This peak cannot be identified in the LBL simulations nor in the baseline TBL simulation without the wall-attached cube.