The Democrats also told Trump to lean hard on Kim's ally China, to ensure it "will do all it can to help secure an agreement and then insist on strict North Korean compliance with such an agreement."
Easing sanctions under a deal would likely need approval from Congress which has passed sanctions on North Korea. Since most legislation needs 60 votes to pass the 100-member Senate and Trump's fellow Republicans hold only 51 seats, that would require Democratic support.
The Democrats' demands include North Korea dismantling and removing every nuclear, chemical and biological weapon, ending production and enrichment of uranium for weapons purposes and permanently dismantling its nuclear weapons infrastructure.
They said North Korea must also agree to suspend ballistic missile tests and commit to robust inspections.
"Now that the meeting will proceed as planned, we want to make sure that the president's desire for a deal with North Korea doesn't saddle the United States, Japan and Korea with a bad deal," Schumer told a conference call with reporters.