WASHINGTON—President Donald Trump said his on-again, off-again summit with North Korean leader Kim Jong Un will proceed in Singapore on June 12 as initially planned, another dramatic turn in a diplomatic saga that has veered from threats of nuclear showdown to talk about peace over a matter of days.

Mr. Trump’s reversal on Friday followed an extraordinary Oval Office meeting with North Korean Gen. Kim Yong Chol, one of Mr. Kim’s top lieutenants and a former spymaster who has been sanctioned by the U.S. Treasury for his role...