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President Trump said this in a press conference on Friday:

I believe [President Obama] would have gone to war with North Korea. I think he was ready to go to war. In fact, he told me he was so close to starting a big war with North Korea.

I’m interested in the part in bold. My question is, is President Trump right that President Obama told him that he was close to starting a war with North Korea?

No, based on this summary of the evidence by the New York Times and other sources:

It is impossible to prove a negative, of course, but nobody who worked
for Mr. Obama has publicly endorsed this assessment, nor have any of
the memoirs that have emerged from his administration disclosed any
serious discussion of military action against North Korea. Several
veterans of the Obama era made a point of publicly disputing Mr.
Trump’s characterization on Friday.

“We were not on the brink of war with North Korea in 2016,” Benjamin
J. Rhodes, Mr. Obama’s deputy national security adviser, wrote on
Twitter.

John Brennan, Mr. Obama’s C.I.A. director, told NBC News, “President
Obama was never on the verge of starting any war with North Korea,
large or small.”

USA Today has a similar take:

"It is absolutely ridiculous to suggest that the Obama administration
was considering anything like that," said Michael Fuchs, who served as
Obama's deputy assistant secretary of State for East Asian and Pacific
affairs. He said Trump was simply "lying" about that exchange.

President Trump has provided no evidence to support his claim. Obama has been silent on the issue. He and the other living American presidents have (at least generally) not responded to Trump's comments about them, as noted in this March 29, 2018 Los Angeles Times story:

Obama expected Trump to keep up his campaign rhetoric and to use Obama
as a “foil to galvanize his base,” especially in moments when Trump
felt the need to boost his political standing, said Josh Earnest,
Obama’s former press secretary and a close advisor. For Obama to
return fire would make it a bigger story.

“Obama engaging Trump has a measurable upside for Trump,” said
Earnest. “But there’s no obvious benefit for the country or, of
course, Obama.”

Trump has said Obama told him this during a November 2016 meeting at the White House. But Trump's initial recollections -- as well as coverage of that meeting at the time, including this Nov. 22, 2016 Wall Street Journal story -- characterize the Obama-Trump interaction as Obama warning Trump about the challenges of dealing with North Korea:

The Obama administration considers North Korea to be the top national
security priority for the incoming administration, a view it has
conveyed to President-elect Donald Trump’s transition team, according
to people familiar with the conversations.

Only in mid-2018 did Trump recast that conversation to say that Obama told him that Obama had been close to going to war with North Korea.

Here's a CNN story from Sept. 26, 2018 addressing and later debunking President Trump's war claims:

"President Obama thought you had to go to war. You know how close he
was to pressing the trigger," Trump said at the press conference,
adding that "not thousands ... millions of people would have been
killed" in what "could have been a world war."

"If I wasn't elected, you'd be in a war," Trump declared, seeking to
bolster his claim by implying that Obama had "essentially" told him so
directly.