To start up a fusion reactor, it is necessary to provide a sufficient amount of tritium from an external device. The fusion DEMO reactor is planned to start up in the 2030s. Herein, methods for supplying the reactor with tritium are discussed. For the initial startup of the fusion reactor, use of a high temperature gas-cooled reactor (HTGR) as a tritium production device has been proposed. So far, the analyses have been focused only on the operation in which fuel is exchanged at stated periods (batch) using the block type HTGR. In this paper, to improve the performance of tritium production, properties of the HTGR are studied from the viewpoint of continuous operation for several conditions. In continuous operation, for example, in the pebble bed type HTGR, it is possible to design an operation that has no time loss for refueling. The pebble bed modular reactor (PBMR) and the gas turbine high temperature reactor of 300 MWe nominal capacity (GTHTR300) are assumed as the calculation and comparison targets, and simulation is made using the continuous-energy Monte Carlo transport code MVP-BURN. It is shown that the continuous operation using the pebble bed type HTGR has almost the same tritium productivity compared with the batch operation using the block type HGTR. The issues for pebble bed type HTGR as a tritium production device are discussed.