We study the relationship between two kinds of higher-order extensions of model checking: HORS model checking, where models are extended to higher-order recursion schemes, and HFL model checking, where the logic is extedned to higher-order fixpoint logic. Those extensions have been independently studied until recently, and the former has been applied to higher-order program verification. We show that there exist (arguably) natural reductions between the two problems. To prove the correctness of the translation from HORS to HFL model checking, we establish a type-based characterization of HFL model checking, which should be of independent interest. The results reveal a close relationship between the two problems, enabling cross-fertilization of the two research threads.