Moon Jae-in could travel to Singapore for the summit depending on the outcome of discussions between the US and North Korean leaders, Seoul’s presidential Blue House said.
Mr Moon and Mr Kim agreed a summit between North Korea and the US must be held as they held a surprise meeting on Saturday at the border village of Panmunjom.
Last week, Mr Trump pulled out of the meeting with Mr Kim, which had been scheduled for 12 June.
He later announced he was reconsidering, and said US and North Korean officials were meeting to work out the details.
A South Korean official told the Yonhap news agency: “We are still waiting to see how they come out, but depending on their outcome, the president could join President Trump and Chairman Kim in Singapore.”
On Sunday, US and North Korean officials met in the border town of Panmunjom, the US State Department said.
Mr Trump tweeted about the developments, saying: ”Our United States team has arrived in North Korea to make arrangements for the Summit between Kim Jong-un and myself.
“I truly believe North Korea has brilliant potential and will be a great economic and financial Nation one day. Kim Jong-un agrees with me on this. It will happen!”
A “pre-advance” team of US officials was also travelling to Singapore to meet with North Koreans there, the White House said.
It comes as Michael Hayden, the former director of the CIA, said he was unsure Mr Trump has “done the homework needed” to negotiate with North Korea about its nuclear programme.
“Kim Jong-un knows his [nuclear] programme in and out,” Mr Hayden told ABC News. “I think he knows what he can concede and what it means and what he cannot concede.
“I don’t know that the president has done the kind of homework” to give him a similar level of knowledge, he added.
In a statement published by the North’s state-run news service on Sunday, Mr Kim “expressed his fixed will on the historic [North Korea]-US summit talks”.
It added that during the inter-Korean meeting on Saturday, both Korean leaders had agreed to “positively cooperate with each other as ever to improve [North Korea]-US relations and establish [a] mechanism for permanent and durable peace”.
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