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 has expressed an "eagerness to meet" with Trump.
She will be the first member of North Korea's ruling family to visit the south since the end of the Korean War.
This extraordinary executive power has been contested throughout American history.
Trump was responding to comments made by North Korean leader Kim Jong Un.
It keeps making outsized threats about attacking the U.S. while ignoring the fact that the U.S. could absorb multiple nuclear missile attacks.