We study spontaneously generated entanglement (SGE) between two identical multilevel atoms in free space via vacuum-induced radiative coupling. We show that the SGE in two-atom systems may initially increase with time but eventually vanishes in the time scale determined by the excited-state lifetime and radiative coupling strength between the two atoms. We demonstrate that steady-state SGE can be established by incoherently pumping the atoms to their excited states. We have shown that an appropriate rate of incoherent pump can help in producing optimal steady-state SGE. The multilevel systems offer us more channels to establish entanglement. The system under consideration can be realized in a tight trap or atoms/ions doped in a solid substrate.