  By applying the island rule proposed recently, we compute the entanglement
entropy of Hawking radiation and study the Page curve for the eternal black
holes in massive gravity. We investigate for both the neutral and charged black
holes which the corresponding results of Schwarzschild and
Reissner-Nordstr\"{o}m black holes are restored in the limit of massless
graviton. We show for the neutral and non-extremal charged black holes that the
island is not formed at the early times of the evaporation and hence the
entanglement entropy increases linearly in time. However, for the extremal
charged black hole, the calculation of the entanglement entropy at the early
times without the island is ill-defined because the metric is divergent at the
curvature singularity. This implies that new physics in the UV region must be
taken into account to make the metric behaving smoothly at the very short
distances. At the late times, with the emergence of one island near the event
horizon, the entanglement entropy is saturated by the Bekenstein-Hawking
entropy of black holes. In addition, we analyze the impact of massive gravity
parameters on the size of island, the entanglement entropy, the Page time, and
the scrambling time in detail.
