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.

**ADDED** I still think the answer still is positive. I think manifolds with tangential structures are understood in terms of Madsen-Tillmann spectra using the Madsen-Tillmann-Weiss map; experts can comment more on this and correct me if this is wrong or vague. In the case of trivialisation of the tangent bundle of $m$ dimensional manifolds, the relative spectrum is $\mathbb{S}^{-m}=\Sigma^{-m}S^0$. The general result of Galatius-Madsen-Tillmann-Weiss provides an interpretation of $\pi_i\Omega^\infty\mathbb{S}^{-m}$ in terms of specific submersions (I guess). Now, the point is that $\pi_i^s\simeq\pi_{i-m}\mathbb{S}^{-m}$ for any $m>0$ and I think again using Freudenthal's theorem one can see the answer is positive.