Extracting interactions among multiple dynamical elements is a fundamental but challenging problem. We show a connection between a phase model and Koopman operator on a vector-valued function space such as a vector-valued reproducing kernel Hilbert space. The phase model gives a low-dimensional description of collective oscillations in interacting dynamical elements, and the Koopman operator is a linear operator describing the time evolution of observables of a dynamical system. By estimating the Koopman operator, we obtain the phase model, which enables us to identify synchronization due to interactions only with given observed time-series data.