Donald Trump says that if he had been president in 2001, his immigration policies would have kept terrorists from attacking the World Trade Center on September 11.

"I am extremely, extremely tough on illegal immigration. I'm extremely tough on people coming into this country," Trump said on Fox News Sunday. "I believe that if I were running things, I doubt those families would have -- I doubt that those people would have been in the country."

The billionaire, in an escalating feud with Republican presidential contender Jeb Bush, continued his assault on the legacy of former President George W. Bush.

Get Breaking News Delivered to Your Inbox

"Jeb said 'we were safe with my brother.' 'We were safe.' Well, the World Trade Center just fell down," Trump continued. "Now, am I trying to blame him? I'm not blaming anybody. But the World Trade Center came down. So when he said, 'we were safe' -- that's not safe. We lost 3,000 people."

And had his immigration proposals been in place at the time, Trump said, "there's a good chance that those people would not have been in our country."

For his part, Bush hasn't appreciated the constant barrage of 9/11 remarks against his brother.

"I don't know why he keeps bringing this up," Bush told CNN Sunday. "It doesn't show that he's a serious person as it relates to being commander-in-chief and being the architect of a foreign policy."

Rather than come across as a legitimate contender for the Oval Office, Bush said, the real estate mogul talks "as though he's still on 'The Apprentice,'" referring to the businessman's reality television show.

"I mean, literally, talking about Syria saying 'ISIS should take out Assad, then Russia should take out ISIS' as though it was some kind of board game and not a serious approach is just - this is just another example of the lack of seriousness," Bush said.

Next week, he added, "Mr. Trump is probably going to say that FDR was around when Japan attacked Pearl Harbor."

"It's what you do after that matters. And that's the sign of leadership," Bush continued. "Does anybody actually blame my brother for the attacks on 9/11? If they do, they're totally marginalized in our society."

The mounting war between the two has played out on Twitter, where Trump first fired at the former Florida governor on Friday for his comments during the Republicans' last debate in California.

.@JebBush, At the debate you said your brother kept us safe- I wanted to be nice & did not mention the WTC came down during his watch, 9/11. — Donald J. Trump (@realDonaldTrump) October 17, 2015

Bush took to social media Saturday to bash Trump over his "absurd comments":

.@RealDonaldTrump's absurd comments again raise questions on whether he possesses the judgment to be President: https://t.co/lLr1SF1jwp — Jeb Bush (@JebBush) October 17, 2015

Another leading Republican presidential contender, retired neurosurgeon Ben Carson, waded into the 9/11 debate, saying he didn't blame former President Bush for the attacks.

When asked on ABC News what he made of Trump's suggestions that Bush should share the blame for the terrorist strikes, Carson said he "would probably ask [Trump] what he meant by that."

"I seriously doubt that he's saying that -- that George W. Bush is to blame for it," Carson said. "I certainly don't think so."