GOFFSTOWN, N.H.—Donald Trump suggested Monday that President Barack Obama may be sympathetic to radical Islamists he said inspired the gunman in this weekend’s Orlando, Fla., nightclub attack, alleging in a television interview the president “doesn’t get it or he gets it better than anybody understands.”

In an address in New Hampshire Monday afternoon, Mr. Trump used a broad brush to attack both the president and presumptive Democratic presidential nominee Hillary Clinton for what he claims are lax immigration laws that contributed to the Florida attack that killed 49 people.

“The bottom line is that the only reason the killer was in America in the first place was because we allowed his family to come here,” Mr. Trump said during his a speech here, referring to the New York-born shooter, Omar Mateen, whose parents emigrated to the U.S. from Afghanistan.

The presumptive Republican nominee’s remarks on Fox News earlier in the day mark the latest escalation in his yearslong campaign to paint Mr. Obama as unqualified for office, either by birth or by virtue of his leadership. Mr. Trump spent much of Mr. Obama’s first term in office alleging the president wasn’t a natural-born U.S. citizen despite Mr. Obama’s official Hawaiian birth certificate.

“Look, we’re led by a man that either is not tough, not smart, or he’s got something else in mind,” Mr. Trump said on Fox News. “And the something else in mind—you know, people can’t believe it. People cannot, they cannot believe that President Obama is acting the way he acts and can’t even mention the words ‘radical Islamic terrorism.’ There’s something going on. It’s inconceivable. There’s something going on.”