Donald Trump renewed warnings about Middle Eastern refugees and other immigrants coming to America on the heels of Sunday’s terror massacre in Orlando, while claiming thousands already are in the country who – like the shooter – are “sick with hate.”

Though the gunman, Omar Mateen, was a U.S. citizen, Trump told Fox News on Monday that the U.S. is allowing others into the country who are “no different than this maniac.”

“This could be the all-time great Trojan horse,” Trump said, of those being admitted from Syria and other nations.

The presumptive Republican presidential nominee has only amplified his warnings about immigration and the terror threat since Sunday’s massacre left 49 dead and even more wounded. At the same time, he’s aggressively criticized both President Obama and likely general election rival Hillary Clinton for not initially using the term “radical Islam” in describing the shooter’s religious background.

“She won’t even utter the words,” Trump said Monday. On Sunday, Trump had said Obama should “step down” for not using the term and Clinton should “get out of this race” if she won’t either.

In remarks that were getting widespread attention Monday, Trump also made a vague statement about Obama interpreted by some as a reference to his sympathies.

“He doesn't get it or, or he gets it better than anybody understands," Trump said. "It's one or the other. And either one is unacceptable." He later said Obama is either not tough and smart or has “something else in mind.”

Obama, speaking in the Oval Office after getting briefed on the investigation by top officials, said the U.S. has to go after terror groups and “hit them hard.” He also said the U.S. needs to make it harder for those who wish to inflict harm to obtain weapons.

Meanwhile, Obama said while the investigation is at the preliminary stages, “We see no clear evidence that he was directed externally.” Rather, Obama said it appears the gunman was inspired by extremist information on the Internet.

Speaking Monday on NBC’s “Today,” Clinton directly rejected Trump's criticism on their approach to terror attacks.

“I think Trump, as usual, is obsessed with name-calling and from my perspective, it matters what we do, not what we say," she said. "It matters that we got bin Laden, not what name we called him. But if he is somehow suggesting I don't call this for what it is, he hasn't been listening.”

Clinton called the attack an “act of terror” on Sunday, as did Obama.

But unlike Obama, who did not use the term "radical Islam," Clinton elaborated on Monday and addressed some of the religious motivations.

“I have clearly said that we face terrorist enemies who use Islam to justify slaughtering innocent people,” she said. “We have to stop them and we will. We have to defeat radical jihadist terrorism, and we will. And to me, radical jihadism, radical Islamism, I think they mean the same thing.

“I'm happy to say either, but that's not the point. … I'm not going to demonize and demagogue and declare war on an entire religion.”

Clinton, who spoke as well on CBS’ “This Morning,” also said she would push for laws that prevent people on America's no-fly list from buying weapons. She also urged tougher restrictions on "weapons of war," like the assault-style rifles used in Orlando and San Bernardino, Calif. – while saying she would set up a federal team dedicated exclusively to addressing "lone wolf" attacks.

Trump later claimed credit on Twitter for Clinton referring to radical Islam.

The attack has only fueled tensions in the 2016 race, as both Trump and Clinton launch into a general election battle after having effectively clinched their respective nominations. Trump has tried to cast Clinton as weak on security issues, while the presumptive Democratic nominee has described Trump as unfit to lead.

Clinton will attend a campaign event in Ohio later Monday, while Trump is scrapping plans to make an anti-Clinton speech and instead delivering an address focused on terrorism, security and immigration Monday afternoon in New Hampshire.

Trump told Fox News he’ll be calling for strength, and calling for intelligence improvements in his speech.

Further, of those in America “sick with hate,” Trump said they need to be reported.

“Nobody’s reporting these people,” he said.

The Associated Press contributed to this report.