1. Suppose $(M, \partial M)$ be an $n$-dimensional manifold with boundary, and suppose $E$ be a $(k-1)$-dimensional subbundle of $\mathrm{T} (\partial M)$. Then, could we always find a $k$-dimensional subbundle $F$ of $\mathrm{T} M$, such that $$ E = F|\partial M \cap\mathrm{T} (\partial M)? $$
So, here is the data : $(M, \partial M, E, F, k)$ are given. Then, what is the obstruction to extend the $E$ to $F$? If $k = n$, there is nothing to do, this is trivial case. In particular, I really be interested in the case of $k = n-1$.
2.(EDITED) This may be solved if the first question is solved. What if the same question is asked if the rank of $F$ is just larger or equal to the rank of $E$($F$ could be $r$-dimensional subbundle and $n \ge r \ge k-1$, don't need to be $r = k$.) and still satisfies the transversal extension condition, $$ E = F|\partial M \cap\mathrm{T} (\partial M)? $$
How could I solve this question? What concepts are needed?
Here is the simplest example when an extension is impossible. Take $M=D^2$, closed 2-disk, $k=1$. Then your condition amounts to finding a line field $F$ on $D^2$ which is transversal to the boundary. Such a line field cannot exist, for instance, since $S^2$ does not admit line fields (as it has nonzero Euler characteristic). Alternatively, you can use a relative form of the Poincare-Hopf theorem. Similar examples exist in higher dimensions, say, when $n=4$ and $k=3$.