Mr Kim's Sunshine Policy led to improved relations with North Korea
North Korea has sent condolences for the death of the former South Korean president, Kim Dae-jung, the North's official news agency has said.
Pyongyang also said it would like to send a delegation to pay respects at Mr Kim's funeral in Seoul.
Relations between the countries have been poor since President Lee Myung-bak took office in the South last year.
But the North has said it wants to ease border restrictions and re-open a joint industrial park near the border.
North Korea's state news agency, KCNA, carried the brief message from the North's leader, Kim Jong-il.
"I express my deep condolences to Mrs Ri Hui Ho and other bereaved family members," he said.
"Though he passed away to our regret, the feats he performed to achieve national reconciliation and realize the desire for reunification will remain long with the nation."
A long-time aide to the former president also said Kim Jong-il had sent condolences and had announced he wanted to send envoys to the funeral.
"The delegation will carry a wreath sent by Chairman Kim Jong-Il," the aide, Park Jie-won, told reporters.
Seoul's unification ministry, which handles cross-border relations, said it had not received word of the delegation from Pyongyang. But President Lee Myung-bak's office said it would not object to the visit.
No date has been set for Mr Kim's funeral.
Kim Dae-jung made the first visit by a South Korean leader to the North in 2000, as part of his "Sunshine Policy" of reconciliation to try to reunite the divided peninsula.
The friendly overtures from the North follow a visit to Pyongyang earlier this month by former US President Bill Clinton.
He held talks with the reclusive Kim Jong-il and secured the release of two American reporters jailed for entering North Korea without permission.
Mr Kim, who died on Tuesday, was being treated for pneumonia.
The former leader had spent his life pursuing democracy and reunification with the North.
He survived several attempts on his life and was awarded the Nobel Peace Prize in 2000.
Kim Dae-jung's date of birth is unclear. According to his presidential website he was born on 6 January 1924, but it is reported that he later changed this to 3 December 1925 to avoid conscription during the Japanese colonial period.
Mr Kim was branded a dangerous radical during South Korea's decades of military rule.
He served as president from 1998 to 2003.
He described the biggest achievement of his presidency as the landmark summit with Kim Jong-il in 2000. It paved the way for reconciliation and earned him a Nobel prize later that year.