Democratic standard-bearer Clinton said last week that she would meet Abdel Fattah al-Sisi, along with leaders from Japan and Ukraine on the sidelines of the UN General Assembly.
She spent more than an hour locked in private discussions with the Egyptian leader, with aides later saying they had discussed issues ranging from counter-terrorism to human rights.
Trump’s meeting with Sisi — the first time the Republican presidential candidate has met a leader from the Muslim world — appeared to be a response to his opponent’s program.
The former reality TV star, who has previously called for a ban on Muslims entering the United States, spoke with Sisi at New York’s La Guardia airport, after flying in from Florida.
Trump told Sisi — who has been much-criticised by rights groups for his intolerance of dissent — that “under a Trump administration, the United States of America will be a loyal friend, not simply an ally, that Egypt can count on.”
Trump has called for “extreme vetting” of refugees entering the United States along with a ban on people from “terrorist nations.”
But he also told Sisi about “his high regard for peace-loving Muslims,” according to a statement released by his campaign.
In an interview with Fox television earlier on Monday, Trump alluded to the impending visit, saying that meetings with other foreign leaders were also on his agenda.
“I don’t want to comment specifically on who but a couple of people are coming over. I’ve already met with a couple,” he said.
As well as a sit-down with Japanese Prime Minister Shinzo Abe, Clinton also met Ukraine’s President Petro Poroshenko, stressing that the US would help Kiev “in the face of Russian aggression” — a dig at Trump, who has repeatedly praised President Vladimir Putin.