Netanyahu was the first leader to have a one-on-one with Trump at the US president's first United Nations General Assembly, an annual week of high-level diplomacy.
The conversation in New York will be dominated by talks of North Korean provocations, the Iran nuclear deal, the crisis in Syria and the debate on climate change.
But Trump was at pains to remind reporters that he has not forgotten his pledge to help negotiate an end to the long dispute between Israel and the Palestinians.
"We're going to be discussing many things; among them, peace between the Palestinians and Israel -- it will be a fantastic achievement," he said, sitting with Netanyahu.
"We are giving it an absolute go. I think there's a good chance that it could happen. Most people would say there's no chance whatsoever," he said.
"I actually think with the capability of Bibi and, frankly, the other side, I really think we have a chance," he insisted.
"I think Israel would like to see it, and I think the Palestinians would like to see it. And I can tell you that the Trump administration would like to see it.
"So we're working very hard on it. We'll see what happens. Historically, people say it can't happen. I say it can happen."
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'Bureaucracy' holding UN back: Trump
Netanyahu thanked Trump for his and the United States' support for Israel, and said he wanted to focus on the Iranian threat and Tehran's growing clout in Syria.
But he also agreed to discuss the "opportunity for peace" between Israel and Palestinians and, in his preferred emphasis, "between Israel and the Arab World."
Trump is due to meet the Palestinian leader, Mahmud Abbas on Wednesday, again at the UN assembly.
Shuttle diplomacy
The Palestinians are keen to clarify US support for a two-state solution to the stand-off with Israel, the focus of international diplomacy since at least the 1990s.
Some members of Netanyahu's government oppose such a deal, and the premier is a defender of Jewish settlement building on occupied land claimed by Palestinians.
Trump's Middle East envoy and son-in-law Jared Kushner and senior aide Jason Greenblatt have been ferrying between leaders from the two sides for months.
But Palestinian officials have become increasingly frustrated by the White House's ambivalent stance.
Trump and Netanyahu were also meeting on the day that, according to a senior Israeli officer, Israel and its US ally opened their first joint missile defense base.
The site, in southern Israel, will fly a US flag and strengthen the region's defenses, said Brigadier General Tzvika Heimowitz, head of Israeli missile defenses.
Trump at the United Nations
US President Donald Trump on Monday warned "bureaucracy" is holding the United Nations back, a barbed first message to an institution he once derided as a talking shop.
Kicking off a frantic week of diplomacy with a panel discussion on UN reform, Trump noted a personal history with the New York-based institution.
He had seen "great potential right across the street" from UN headquarters, Trump said, referencing his decision to build the 72-floor residential skyscraper Trump World Tower nearby.
"To be honest with you, and it was only for the reason that the United Nations was here that that turned out to be such a successful project," the businessman-turned-president told delegates.
But Trump warned that as chief executive of the United States -- a founding member of the UN and its biggest financial contributor -- he wants a better return on his investment.
"The United Nations was founded on truly noble goals" he said, adding that while progress has been made "in recent years the United Nations has not reached its full potential, because of bureaucracy and mismanagement."
"While the United Nations on a regular budget has increased by 140 percent, and its staff has more than doubled since 2000, we are not seeing the results in line with this investment."
He called on the institution to "focus on results," a call that was echoed by UN Secretary General Antonio Guterres, who said a reformed UN needs to focus "more on people and less on process."
Trump, once disparaged the world body as a "club" for "people to get together, talk and have a good time" and his administration has threatened deep cuts to UN funding.
That includes a $600 million cut to the peacekeeping budget, which critics warn will be devastating but which is music to Trump supporters' ears.
"I think the main message is 'Make the United Nations Great.' Not again. 'Make the United Nations Great'' Trump said after his remarks.
"The United Nations has tremendous potential and we'll see how it works out."
Although president Franklin Roosevelt was instrumental to midwifing the United Nations into existence, the US public has proven more skeptical.
Many of Trump's closest advisors and donors see the 193-member body as a check on US power and drain on American resources.
The United States pays 22 percent of the UN's $5.4 billion core budget and 28.5 percent of the $7.3 billion peacekeeping budget.
According to Gallup 60 percent of Americans think the United Nations is doing a poor job.
The Trump administration's effort to cut costs has been met coolly by other powers, most notably Russia and China, who were not among the 128 member states who signed on to a US call for reform.
