In this paper, we show that if the non-constant rational functions f1, …, fn∈ K(x) over a number field K cannot multiplicatively generate a power of a linear fractional function, then there are only finitely many elements α∈ K such that f1(α),…,fn(α) are multiplicatively dependent modulo some subset `close' (with respect to the Weil height) to the division group of a finitely generated multiplicative subgroup of K. This improves some previous results.