Specifically, the GOP front-runner thinks Rafael Cruz may have had some sort of connection to the assassination of John F. Kennedy.
"I mean, what was he doing — what was he doing with Lee Harvey Oswald shortly before the death? Before the shooting? It's horrible," Trump speculated.
9 facts you should know about Ted Cruz:
9 Facts you should know about Ted Cruz
Trump throws another dagger, says Cruz's father may be connected to JFK's death
1) His legal name is Rafael Edward Cruz.
(Photo: PAUL J. RICHARDS/AFP/Getty Images)
2) His wife Heidi has worked at investment banking firm Goldman Sachs. The company told CNN Monday she will go on unpaid leave for the duration of his campaign. They met while they worked on George W. Bush's 2000 presidential campaign.
(AP Photo/Andrew Harnik)
3) He won his Senate seat in 2010 without ever having been elected to public office before. Prior to that he had been appointed to the office of the Solicitor General in Texas.
(Photo by Mark Wilson/Getty Images)
4) He had a minor brush with the law in 1987 when he received a ticket for underage possession of alcohol as a senior in high school.
(Photo by: Lloyd Bishop/NBC/NBCU Photo Bank via Getty Images)
5) He has two Ivy League degrees: an undergraduate degree from Princeton, and a law degree from Harvard.
(Photo by Chip Somodevilla/Getty Images)
6) He has kept a painting of himself in his office -- a picture of him as a 32-year-old arguing a case before the Supreme Court.
(AP Photo/Harry Cabluck)
7) He played a significant role in the government shutdown of 2013, leading a chorus of Republicans who refused to vote for any plan that kept the federal government running that did not also defund Obamacare. Cruz spend nearly 24 straight hours defending his position, including at one point reading the Dr. Seuss classic "Green Eggs and Ham."
(AP Photo/Senate TV)
8) His father (left) fled Cuba for the United States, worked in the oil industry and eventually became a pastor. He has made headlines for somewhat inflammatory statements, including telling an audience that President Obama should be sent "back to Kenya."
(Ron T. Ennis/Fort Worth Star-Telegram/MCT via Getty Images)
9) He doesn't believe in climate change, an issue many Democrats have lampooned him for, in part because he leads the Senate's Space, Science, and Competitiveness Committee which oversees NASA. During a recent appearance on "Late Night with Seth Meyers" Cruz said "Debates on this should follow science and should follow data. Many of the alarmists on global warming, they’ve got a problem because the science doesn’t back them up."
(Photo by Scott Olson/Getty Images)
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The National Enquirer tabloid ran a story in April alleging the same thing, though with shaky evidence at best — a photo of Lee Harvey Oswald with an unidentified man the tabloid suggests is the elder Cruz.
But it's worth noting that even the experts the National Enquirer hired to analyze the picture wouldn't say the man is Rafael Cruz — just, basically, that it looks like him.
Cruz's campaign told McClatchy the claim was "another garbage story in a tabloid full of garbage." The National Enquirer, by the way, endorsed Trump last month.