Singular moduli, or special points, defined as complex numbers which are the images of quadratic points under the $j$-invariant, form one of the fundamental objects in number theory. In 2022, J.~S. Pila and J. Tsimerman [J. Eur. Math. Soc. (JEMS) {\bf 24} (2022), no.~9, 3161--3182; MR4422210] established a very interesting Zilber-Pink type result for mixed Shimura varieties, which deals with the multiplicative independence property of singular moduli. More precisely, they proved that for $n\geq 1$, there are only finitely many $n$-tuples $(\sigma_1,\dots,\sigma_n)$ of singular moduli such that the set $\{\sigma_1,\dots,\sigma_n\}$ is multiplicatively dependent but no proper subset of $\{\sigma_1,\dots,\sigma_n\}$ is multiplicatively dependent. \par In this paper, the author generalizes the above theorem of Pila and Tsimerman to a much broader class of modular functions, i.e., those satisfying the following divisor condition: \par Divisor Condition. Let $f\:\Bbb{H}\to\Bbb{C}$ be a non-constant modular function. Let $w_1,\dots,w_r$ be all the zeros and poles of $f$ contained in the standard fundamental domain of the $j$-invariant, enumerated so that their imaginary parts are increasing. Then we say $f$ satisfies the divisor condition if for any $s\in(0,1)$, we have that $w_r+s$ is neither a zero nor a pole of $f$. \par Notice that the notion of singular moduli generalizes naturally for any modular functions $f$: we call a complex number $f$-special if it has the form $f(\tau)$ for a quadratic point $\tau\in\Bbb{H}$. Then the main theorem of this paper reads: \par Theorem. Let $f\:\Bbb{H}\to\Bbb{C}$ be a non-constant modular function satisfying the divisor condition. Then, for $n\geq 1$, there are only finitely many $n$-tuples $(\sigma_1,\dots,\sigma_n)$ of $f$@-special points such that the set $\{\sigma_1,\dots,\sigma_n\}$ is multiplicatively dependent but no proper subsets of $\{\sigma_1,\dots,\sigma_n\}$ are multiplicatively independent. \par It is worth noting that while in the original approach of Pila and Tsimerman they proved a functional transcendence theorem by using some arguments which do not readily generalize to other modular functions, the author of this paper gives a new proof of this transcendence theorem by applying only elementary properties of $j$, which enables one to easily generalize to a wider class of modular functions. \par Moreover, we emphasize that the divisor condition does not come from nowhere. In fact, the author verifies this condition for modular functions which appear naturally as Borcherds lifts of certain weakly holomorphic modular functions. \par Finally, the author explicitly explains how the results of this paper fit the framework of the very technical Zilber-Pink conjecture for mixed Shimura varieties.