We develop an interior-point method for nonsmooth regularized bound-constrained optimization problems. Our method consists of iteratively solving a sequence of unconstrained nonsmooth barrier subproblems. We use a variant of the proximal quasi-Newton trust-region algorithm TR of Aravkin et al. (2022) to solve the barrier subproblems, with additional assumptions inspired from well-known smooth interior-point trust-region methods. We show global convergence of our algorithm with respect to the criticality measure of Aravkin et al. (2022). Under an additional assumption linked to the convexity of the nonsmooth term in the objective, we present an alternative interior-point algorithm with a slightly modified criticality measure, which performs better in practice. Numerical experiments show that our algorithm performs better than the trust-region method TR, the trust-region method with diagonal hessian approximations TRDH of Leconte and Orban (2023) and the quadratic regularization method R2 of Aravkin et al. (2022) for two out of four tested bound-constrained problems. On the first two problems, RIPM and RIPMDH obtain smaller objective values than the other solvers using fewer objective and gradient evaluations. On the two other problems, our algorithm performs similarly to TR, R2 and TRDH.