  The possibility for the existence of unstable bound states of the S11 nucleon
resonance N$^*$(1535) and nuclei is investigated. These quasibound states are
speculated to be closely related to the existence of the quasibound states of
the eta mesons and nuclei. Within a simple model for the N N$^*$ interaction
involving a pion and eta meson exchange, N$^*$-nucleus potentials for N*-$^3$He
and N*-$^{24}$Mg are evaluated and found to be of a Woods-Saxon like form which
supports two to three bound states. In case of N*-$^3$He, one state bound by
only a few keV and another by 4 MeV is found. The results are however quite
sensitive to the N N$^*$ $\pi$ and N N$^*$ $\eta$ vertex parameters. A rough
estimate of the width of these states, based on the mean free path of the
exchanged mesons in the nuclei leads to very broad states with $\Gamma \sim$ 80
and 110 MeV for N*-$^3$He and N*-$^{24}$Mg respectively.
