Trump again denied he or his campaign had anything to do with the National Enquirer report. Cruz has blamed the report on Trump and his "henchmen" while forcefully denying it.
"I had nothing to do with it. The campaign had absolutely nothing to do with it," Trump told ABC's Jonathan Karl on "This Week" Sunday. "I have over control over the National Enquirer."
In an appearance on Fox News, Cruz said he doesn't believe that Trump had no involvement in the story's publication.
"This story is garbage. It is tabloid smear. It came from Donald Trump and his henchmen," Cruz said, adding the report is "indicative of just how low Donald Trump will go."
Trump said it was "disgraceful" that Cruz had blamed him for the story.
Trump also declined to say whether he condemned the National Enquirer report or whether he thought it was accurate.
"I don’t care. The National Enquirer did a story. It was their story -- it wasn’t my story," he said. "It was about Ted Cruz. I have no idea if it's right or not. They actually have a very good record of being right. But I have absolutely no idea. Frankly, I said, I hope it's not right.”
The National Enquirer allegations have not been confirmed by ABC News.
Trump had less to say about his threat to "spill the beans" on Cruz's wife, Heidi. He also defended promoting a side-by-side image of Heidi Cruz and his wife Melania that said: "No need to spill the beans. The images are worth a thousand words."
"It was a retweet that somebody sent me," Trump said.
"What he did was far worse," he added, referring to the super PAC ad featuring his wife.
On Fox News, Cruz said Trump began criticizing his wife to divert attention from remarks he made about possibly withdrawing the United States from NATO.
"Donald did what he always does -- he tried to find a way to change the subject," said Cruz. "He hasn't campaigned for a week. He's been hiding in Trump Tower, but late at night he sends tweets attacking my wife, attacking Heidi. It is inappropriate, wrong and it is frankly disgusting to see a candidate attacking the spouse of another."
Cruz called the ad that used Melania's photo "deplorable" and denied having anything to do with it.
"The fact that someone else who I don't know and have no involvement with used a deplorable ad," he said. "He used that as an excuse to go after my wife Heidi, to go after my family, to attack her directly, and not only did he threaten her directly."