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We consider a directed polymer of length $N$ interacting with a linear interface. The monomers carry i.i.d. random charges $(ω_i)_{i=1}^N$ taking values in $\mathbb{R}$ with mean zero and variance one. Each monomer $i$ contributes an energy $(βω_i-h)φ(S_i)$ to the interaction Hamiltonian, where $S_i \in \mathbb{Z}$ is the height of monomer $i$ with respect to the interface, $φ: \mathbb{Z} \to [0,\infty)$ is the interaction potential, $β\in [0,\infty)$ is the inverse temperature, and $h \in \mathbb{R}$ is the charge bias parameter. The configurations of the polymer are weighted according to the Gibbs measure associated with the interaction Hamiltonian, where the reference measure is given by a Markov chain on $\mathbb{Z}$. We study both the quenched and the annealed free energy per monomer in the limit as $N\to\infty$. We show that each exhibits a phase transition along a critical curve in the $(β, h)$-plane, separating a localized phase (where the polymer stays close to the interface) from a delocalized phase (where the polymer wanders away from the interface). We derive variational formulas for the critical curves and we obtain upper and lower bounds on the quenched critical curve in terms of the annealed critical curve. In addition, for the special case where the reference measure is given by a Bessel random walk, we derive the scaling limit of the annealed free energy as $β, h \downarrow 0$ in three different regimes for the tail exponent of $φ$.