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Cruz reportedly suggested that Trump's attacks came in response to a new poll that showed him losing in a head-to-head competition with the senator.
"That has got to drive him nuts, and I imagine it sent him out of bed this morning tweeting and tweeting and tweeting," Cruz said, according to The Texas Tribune and other outlets.
Cruz then reportedly said Trump's behavior wouldn't be appropriate for a "commander in chief."
"I think in terms of a commander in chief, we ought to have someone who isn't springing out of bed to tweet in a frantic response to the latest polls," he said. "I think the American people is looking for a commander in chief who is stable and steady and a calm hand to keep this country safe," Cruz added.
The New York Times further reported that Cruz described Trump as having a "lot of nervous energy."
"And for whatever reason, Donald doesn't react well when he's going down in the polls," he added.
Trump easily leads Cruz in national polls, but Cruz has surged in recent weeks to roughly tie Trump in polls of the February 1 Iowa caucus, the first sate to weigh in on the presidential primary.
Trump and Cruz were once campaign-trail allies, with Cruz in particular taking painstaking efforts to avoid criticizing the front-runner. But two weeks ago, Trump started questioning Cruz's eligibility to be president on a daily basis, and their détente melted away.
On Saturday morning and afternoon, Trump got especially aggressive on Twitter. For eight hours, Trump attacked Cruz for not reporting two bank loans he used to finance his 2012 Senate campaign, for being born in Canada, and for bashing Trump's "New York Values."
In one particularly provocative Twitter post, Trump posted a photo of the ruins of the September 11, 2001, terror attacks and asked if that was the same New York that Cruz was knocking.
Apparently in response to Trump, Cruz campaign spokesman Rick Tyler tweeted campaign donations that the "real Donald Trump" had given to New York Democratic politicians over the years:
The Real Donald Trump gave $14,100 to NYC liberal Charlie Rangel #NYValues
- Rick Tyler (@rickwtyler) January 16, 2016
The Real Donald Trump gave $8,900 to NYC liberal Democrat Chuck Schumer #NYValues
- Rick Tyler (@rickwtyler) January 16, 2016
The Real Donald Trump gave $4,300 to NYC liberal Democrat Anthony Weiner #NYValues
- Rick Tyler (@rickwtyler) January 16, 2016