In terms of homotopy theory, the questions is that when an element of $\pi_n^s$ pulls back to an element of $\pi_{n+i}S^i$ for some $i$ and the answer to this question is positive by Freudenthal's suspension theorem as there is an epimorphism 
$$\pi_{2n+1}S^{n+1}\longrightarrow \pi_n^s$$
unless you put more restrictions, e.g. ask for some specific $i$. The point is that there could be two different pull backs whose normal bundles are not isomorphic, only stably isomorphic. An example is given by $\eta_3\in\pi_8^3$ which equals to $\eta\sigma=\sigma\eta$ in $\pi_*^s$. However, one can do some unstable computations to show that one pulls back one step further than the other. Hence, as unstable elements they are not the same really, but map to the same element. I think it could be interesting to sort this out in terms of bordism theory and I don't know if such specific examples are considered somewhere in the literature.