The initial design study of 28 GHz/35 GHz dual frequency gyrotron is presented. The design is based on 28 GHz 1 MW gyrotron which we have developed. And it is studied whether 35 GHz oscillation is possible or not using the present 28 GHz gyrotron cavity. From the numerical simulation results, the cavity can form 35.45 GHz TE9,4 mode oscillation of over 1 MW oscillation power with high efficiency. However, in the present mode converter and transmission mirror system, the radiated RF power through the output window is only about 70 % of the 35.45 GHz oscillation power because of the radiation angle difference of 15.85°. For more high efficiency operation for 35 GHz, we will re-design the cavity in which there are good candidate modes such that there is no radiation angle difference.