UPDATE: Donald Trump's campaign issued a statement Tuesday, clarifying his remarks.

Donald Trump seemed to make a strange suggestion Monday that President Obama was somehow connected to the Orlando nightclub massacre that claimed the lives of 49 people early Sunday morning.

The indirect accusation came in an interview with Fox News Channel's "Fox and Friends" Monday.

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"[W]e're led by a man that either is not tough, not smart, or he's got something else in mind," he said. And he went on to say, "[T]he something else in mind, you know, people can't believe it," Trump said.

He went on to say twice that there was "something going on."

"People cannot -- they cannot believe that President Obama is acting the way he acts and can't even mention the words radical Islamic terrorism," Trump told Fox News. "There's something going on. It's inconceivable. There's something going on."

He made similarly ominous and unsubstantiated suggestions on NBC's "Today Show" that Mr. Obama was somehow involved in the attacks, saying that maybe the president "doesn't want to get it" or "he gets it better than anybody understands," he said. "He doesn't get it or he gets it better than anybody understands."

But then, asked to elaborate on the implicit accusation, Trump backed off a little, accusing the president of mere incompetence, rather than malicious intent, though he suggested that unnamed others thought there was a connection.

"I happen to think that he just doesn't know what he's doing," Trump said, "but there are many people that think he doesn't want to get it. He doesn't want to see what's really happening."

On Sunday, Trump said that Obama should step down as president because he refuses to say "radical Islamic terrorism," and warned on "CBS This Morning" on Monday that more Orlando-style attacks will continue to occur if Mr. Obama doesn't start talking about "radical Islamic terrorism."

"Believe me, all I want is safety, I want safety for this country," Trump told "CBS This Morning" in a phone interview. "What happened yesterday will happen many times over with a president like Obama that doesn't even want to use the term 'radical Islamic terrorism.'"

Trump said Sunday that the massacre was proof that the U.S. government should implement his Muslim ban. His proposal, however, would block Muslims from entering the U.S., while the shooter, Omar Mateen, was born in New York to Afghan-born parents.

The president addressed the nation about the attack on Sunday afternoon, condemning the shooting at the gay nightclub in Orlando that left 49 people dead and 53 more wounded.

Mr. Obama said that while the investigation is still underway, it is clear the shooting was an "act of terror and an act of hate." Mateen had pledged allegiance to ISIS in a phone call to 911 during the attack.

The president received a briefing update Monday morning in the Oval Office from FBI Director James Comey, Homeland Security Secretary Jeh Johnson and other administration officials.

CBS News' Sopan Deb contributed to this report.