TOKYO—Substantial damage to the fuel cores at two additional reactors of Japan's Fukushima Daiichi nuclear complex has taken place, operator Tokyo Electric Power Co. said Sunday, further complicating the already daunting task of bringing them to a safe shutdown while avoiding the release of high levels of radioactivity. The revelation followed an acknowledgment on Thursday that a similar meltdown of the core took place at unit No. 1.

Workers also found that the No. 1 unit's reactor building is flooded in the basement, reinforcing...