For some families of distributions, there is a relationship between the two.

The result is not from this paper, but they give the clearest exposition: Otto, Villani, 2000, Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality.

The quantity you are interested in, they define as the relative Fisher information in equation 8.

In definition 1, they give the caracterization of a measure $q$ satisyfing a logarithmic sobolev inequality with constant $\rho$ as satisfying the inequality you seek with constant $\rho$

$$ KL(p, q) \leq \frac{1}{2\rho} I(p,q) $$

The key ingredient we now need is: what families are $LSI(\rho)$. That's stated in theorem 2, due Bakry and Emery: strongly log-concave distributions are $LSI(\rho)$.