Pyongyang argued that a meeting between the family members and recent North Korea defectors would expose the "fiction" that they had escaped to South Korea voluntarily.
The 12 women, working as staff in a North Korean restaurant in China, arrived in the South earlier this month, along with their manager.
Seoul said they had planned their group defection together, while the North insisted they were tricked into defecting by South Korean spies who effectively "kidnapped" them with the connivance of the manager.
Seoul's Unification Ministry, which handles cross-border affairs, swiftly rejected the idea and stressed again that the 12 women had defected voluntarily.
In today's statement, the North Korean Red Cross said it would not take no for an answer.
"The families of the abductees are eagerly asking for face-to-face contact with their daughters as they were forced to part," the statement said.
"At their earnest requests, our side again seriously notifies your side of our decision to send them to Seoul via Panmunjom," it added.
The statement called on the South Korean Red Cross to take "immediate technical measures" to allow the relatives to cross the border and travel to Seoul.
The Unification Ministry in Seoul said it was unaware of any official notice being received.
It is extremely unlikely that the South would allow such a meeting to take place, on the grounds that it would be exploited by the North for propaganda purposes.
Nearly 30,000 North Koreans have fled poverty and repression at home to settle in the capitalist South.
But group defections are rare, especially by staff who work in the North Korea-themed restaurants overseas that are a key source of hard currency for the regime in Pyongyang.