EDITED: I'm not sure why "singular torus" appears in the header, since in the Borel-Chevalley structure theory this refers to a torus which lies in infinitely many Borel subgroups (e.g., the identity component in the kernel of a root).
Anyway, the foundational paper by Borel-Tits Groupes reductifs (IHES Publ. Math., 1965) includes among other things a description of all closed subgroups of a connected semisimple (or reductive) group $G$ which contain maximal tori of $G$. Here it's convenient to reserve the letter $T$ for such a maximal torus, while $S$ can be any subtorus. Now the centralizer $Z_G(S)$H:=Z_G(S)$ is connected, reductive, and contains $T$; more precisely, it is generated by $T$ along with some pairs of root subgroups centralizing $S$ (for pairs of positive and negative roots). In turn, the normalizer $N_G(S)$ is generated by $Z_G(S)$ along with some representatives roots) and part of the Weyl group $W$ of $G$ relative to $T$. This forces Then the quotient Weyl group of $H$ relative to be finite and in fact $T$ is naturally isomorphic to a subgroup of $W$.
[Along the way one relies on the elementary "rigidity" theorem for tori, which shows that the "Weyl group" is always finite.]
This part of the structure theory is summarized briefly in Section 2.1 of my 1995 AMS book Conjugacy Classes in Semisimple Algebraic Groups (with references to Borel-Tits).
P.S. The terminology "Weyl group" here threw me off, so I was trying to answer a different question. The answer to the original question is basically no. Given a connected semisimple group $G$ (say with a maximal torus $T$, where the Weyl group of $G$ relative to $T$ is denoted $W$), the finite group $N_G(S)/Z_G(S)$ for an arbitrary subtorus $S$ of $T$ usually has no direct relationship with $W$. This group is always finite (rigidity of tori), but shouldn't be thought of as a "Weyl group". That notion is usually defined for a pair consisting of an algebraic group together with one of its maximal tori. For instance, the connected reductive group $H:=Z_G(S)$ has $T$ as a maximal torus and then the Weyl group of $H$ relative to $T$ is a subgroup of $W$ (as indicated in my corrected version above). But this is not the finite group in question here. As Florian indicates, in special cases there may still be an indirect connection between the group $N_G(S)/Z_G(S)$ and $W$.
