In an interview on "The Howie Carr Show"
on Tuesday in New Hampshire, Cruz said Trump "may shift in his new rallies to playing 'New York New York,' because you know Donald comes from New York and he embodies New York values."
Cruz's recent polling surge has thrust him into a two-man race with Trump in Iowa while he closes the gap with the GOP front-runner nationally -- and as the race has tightened, the rhetoric has escalated.
Cruz also appeared on "The Kelly File" on Fox News on Tuesday night, and when asked about his "New York" remarks, he repeated that his campaign's success has "really rattled Donald" and cracked, "The rest of the country knows what New York values are."
Trump has attacked his rival with increasing frequency as Cruz's standing in the Republican primary has raised, questioning Cruz's eligibility to serve as president because of his birth in Canada and hitting the senator from Texas over his opposition to ethanol subsidies, an important issue in Iowa
, the first caucus state.
On Wednesday, the day after the interview, Trump continued to knock Cruz over his qualifications, tweeting, "Sadly, there is no way that @TedCruz can continue running in the Republican primary unless he can erase doubt on eligibility. Dems will sue!"
But despite Cruz's insistence on Howie Carr's show that he doesn't "intend to respond or to reciprocate" to the attacks or "go into the mud," the firebrand senator didn't pass up any opportunities to hit back at Trump during his interview.
Asked on the show about Hillary Clinton's comments that Trump "was basically a Democrat before he was a Republican," Cruz noted, "I would say Hillary would know well how to identify Democrats."
"She and Donald know each other well," he went on, "and I do think it's interesting that Hillary Clinton's key supporters are doing everything they can to echo Donald's attacks on me."
Cruz also said polls show that while Trump would lose a hypothetical matchup with Clinton "by a pretty big margin," he would beat her.
Later in the interview, Carr brought up remarks Trump made about the lack of evangelicals from Cuba, where Cruz's father was born.
Cruz replied, "Well listen, anytime someone is attacking your faith, that starts to suggest they're getting really nervous about what's happening in the race."