President Obama will renew his appeal for world leaders to accept more refugees in his final appearance at the United Nations on Tuesday, even as a series of suspected Islamist terrorist attacks in the U.S. raised fresh questions about the administration’s plan for welcoming immigrants from Syria.

The president will convene a summit at U.N. headquarters in New York City, seeking agreements from countries to double the number of refugees resettled each year. To show his commitment, Mr. Obama plans to raise the U.S. limit on refugees next year to 110,000, up from 85,000.

As he prepared to make his international appeal, however, Mr. Obama faced renewed criticism at home from the campaign of Republican presidential nominee Donald Trump over the administration’s immigration and national security policies after suspected terrorist attacks this weekend in New York, New Jersey and Minnesota.

Mr. Trump said the series of attacks “should be a wake-up call for every American” that the U.S. needs tougher immigration standards.

“We need to get smart and get tough fast so that this weekend’s attacks do not become the new normal here as it has in Europe and other parts of the world,” Mr. Trump said in a statement. “I will bring an end to these senseless acts of violence. We will not allow political correctness and soft-on-terror, soft-on-crime policies to threaten our security and our lives.”

Mr. Trump cited an internal Homeland Security Department audit Monday showing that the U.S. mistakenly granted citizenship to at least 858 immigrants from “special interest countries” that present national security concerns or high rates of immigration fraud. He said the report “puts this weekend’s attacks in a broader perspective.”

“The safety and security of the homeland must be the overriding objective of our leaders when it comes to our immigration policy,” Mr. Trump said. “That’s why I’ve proposed extreme vetting for immigrants from troubled parts of the world where terrorists live and train, and oppose Hillary Clinton’s 550 percent increase in the number of refugees from the conflict in Syria.”

Mr. Obama took a cautious approach to the attacks, refusing to characterize the New York-area bombings as acts of terrorism. Hours later, New York Mayor Bill de Blasio blamed the bombings on terrorism. The president scolded the media for reporting suspected terrorism links as the probe was unfolding.

“I would ask that the press try to refrain from getting out ahead of the investigation,” Mr. Obama said. “It does not help if false reports or incomplete information is out there.”

Even before he spoke, however, police in New Jersey captured bombing suspect Ahmad Khan Rahami, a naturalized U.S. citizen from Afghanistan, during a shootout in Linden, New Jersey. Mr. Rahami is suspected in the explosion Saturday that injured 29 people in Manhattan and the pipe-bomb explosion at a charity race Saturday in Seaside Park, New Jersey, where no one was injured.

Mr. Obama later called the arrest of Mr. Rahami “outstanding police work.” He spoke on the phone with officers injured in the shootout and with the officer who killed the knife-wielding attacker in Minnesota.

Although the president didn’t characterize the explosions in New York and Seaside Park as acts of terrorism, he said the Islamic State group “is instigating a lot of people over the internet to carry out attacks.”

“They are trying to hurt innocent people, but they are also trying to instill fear in all of us,” Mr. Obama said. “We all have a role to play as citizens in making sure we don’t succumb to that fear.”

The administration also came under fire Monday for a comment by White House press secretary Josh Earnest, who said the U.S. is in a “narrative fight” with the Islamic State. He cautioned against portraying the fight as a battle between Islam and the West, saying that narrative helps extremists.

Mr. Trump rejected that argument.

“They are wrong to say that we’re in a fight about ‘narratives,’” the Republican nominee said. “These terrorists pose an existential threat to our country, our values and our way of life. When I am president, terrorists like today’s suspect in the New York and New Jersey bombings, Ahmad Khan Rahami, and [the] knife-wielding ISIS sympathizer in Minnesota, Dahir Adan, will be stopped. We will not look the other way.”

Police say Adan, 22, made references to Allah as he stabbed nine people Saturday in a mall in St. Cloud, Minnesota. An off-duty police officer shot him dead.

Adan was described as a member of a Somali family who was born in Kenya.

Before the weekend attacks, the U.S. homeland was the target of 89 Islamist terrorist plots and attacks since 9/11, and 25 since the start of 2015, said David Inserra, a homeland security specialist at the conservative Heritage Foundation.

Also on Monday, Mr. Obama met in New York with Iraqi Prime Minister Haider al-Abadi and predicted a “tough fight” by the U.S.-led military coalition to dislodge Islamic State militants from their stronghold in Mosul.

“We feel confident that we will be in a position to move forward fairly rapidly,” Mr. Obama said. “This is going to be hard, this is going to be challenging. We are going to be asking Congress and other countries to step up to their effort.”

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