Cruz's communications director, Rick Tyler, used a pun to dismiss Trump's attacks.
"There is reason 'Trumped up' means phony," Tyler told Business Insider in an email. "He should look hard in the mirror."
Click through some of Donald Trump's most controversial Tweets:
Craziest Donald Trump Tweets
Ted Cruz's campaign had a snappy response to Donald Trump's lawsuit threat
I told you @TIME Magazine would never pick me as person of the year despite being the big favorite They picked person who is ruining Germany
I would like to wish everyone, including all haters and losers (of which, sadly, there are many) a truly happy and enjoyable Memorial Day!
Just made the point at #NCGOPcon that "we have to protect our border & I think everyone here knows, nobody can build a wall like Trump!"
My twitter has become so powerful that I can actually make my enemies tell the truth.
China's Communist Party has now publicly praised Obama's reelection. They have never had it so good. Will own America soon.
While Jon Stewart is a joke, not very bright and totally overrated, some losers and haters will miss him & his dumb clown humor. Too bad!
Why is Obama playing basketball today? That is why our country is in trouble!
An 'extremely credible source' has called my office and told me that @BarackObama's birth certificate is a fraud.
Remember, I said Derek don't sell your Trump World Tower apartment...its been lucky for you. The day after he sold it, he broke his foot.
I never fall for scams. I am the only person who immediately walked out of my ‘Ali G’ interview
.@cher--I don’t wear a “rug”—it’s mine. And I promise not to talk about your massive plastic surgeries that didn’t work.
The concept of global warming was created by and for the Chinese in order to make U.S. manufacturing non-competitive.
.@katyperry is no bargain but I don't like John Mayer--he dates and tells--be careful Katy (just watch!).
.@BetteMidler talks about my hair but I'm not allowed to talk about her ugly face or body --- so I won't. Is this a double standard?
It's Thursday. How much money did Barack Obama waste today on crony green energy projects?
Truly weird Senator Rand Paul of Kentucky reminds me of a spoiled brat without a properly functioning brain. He was terrible at DEBATE!
Hey @realDonaldTrump - In case you're looking for that tweet you deleted .. I got it right here. http://t.co/a6F98q5xGU
This very expensive GLOBAL WARMING bullshit has got to stop. Our planet is freezing, record low temps,and our GW scientists are stuck in ice
Everyone knows I am right that Robert Pattinson should dump Kristen Stewart. In a couple of years, he will thank me. Be smart, Robert.
26,000 unreported sexual assults in the military-only 238 convictions. What did these geniuses expect when they put men & women together?
Sorry losers and haters, but my I.Q. is one of the highest -and you all know it! Please don't feel so stupid or insecure,it's not your fault
How amazing, the State Health Director who verified copies of Obama’s “birth certificate” died in plane crash today. All others lived
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The day before, Trump released a statement raging about Cruz's supposedly "fraudulent" campaign actions.
Trump has accused Cruz of running robocalls saying that Trump was not going to run in the upcoming South Carolina primary.
During a Monday press conference, Trump said the calls were "much more severe than anything that you had" during the Iowa caucuses. This was a reference to the Cruz campaign implying to its Iowa precinct captains that retired neurosurgeon Ben Carson was dropping out of the race.
The mogul has also alleged that Cruz cannot be president because he was born in Canada. Although most legal experts believe Cruz meets the Constitution's requirement that presidents be "natural-born" citizens because his mother was a US citizen at the time of his birth, Trump has repeatedly cast doubt on Cruz's eligibility.
Trump previously said on Friday that he had "standing" to sue if Cruz didn't "clean up his act, stop cheating, & doing negative ads." Trump said at the press conference that he already hired a lawyer.
"If Ted is going to continue to lie with such desperation, I have no choice but to fight back," Trump said.