The dynamical characteristics of two linearly coupling solitons with a localized impurity in Bose-Einstein condensates are investigated by direct numerical simulations of the Gross-Pitaevskii equation with split-step Fourier method. It is found that the evolution of solitons depends on the impurity strength and linearly coupling parameters of the system, and there exist exchange, self-trapping and Josephson oscillator with the increasing of linearly coupling parameters. The effect of impurity strength of potential well on the evolution of solitons is little when some given values are taken.