Officials in Pyongyang sent the warning Thursday.
North Korea said it would "strike mercilessly without notice" in response to rallies in Seoul this week on the second anniversary of the death of former North leader Kim Jong Il, CNN reported.
Demonstrators burned effigies of Kim and protested the North's authoritarian rule and human rights abuses, the Yonhap News Agency reported Friday.
The fax message said the North would strike if "the provocation against our highest dignity is to be repeated in the downtown of Seoul."
Kim Min-seok, spokesman for South Korea's Ministry of National Defense, said if the North carries out its threat "we will firmly retaliate."
North Korea frequently makes threats against the South and it is a common practice for rallies to be held in the South during North Korean festivals and anniversaries, the report said.