South Korea says they have "nothing to confirm" regarding the 36-year-old's health.
The two leaders shook hands on the steps of a hotel in Sentosa, Singapore.
The highly anticipated meeting is expected to happen on Tuesday.
"We’ll be meeting June 12 in Singapore. You people will have to be in Singapore."
He wrote a letter to Kim, which was sent to reporters.
Kim criticized the White House for pushing the narrative that North Korea had chosen to denuclearize because of U.S. sanctions and pressure.
Welcome to 2018.
It also plans to close a test site.
Kim has expressed an "eagerness to meet" with Trump.
The pariah state sent 229 cheerleaders to the 2018 Winter Olympics in PyeongChang.