"I don't think he has the right temperament," Trump said of Cruz on "Fox News Sunday." "I don't think he's got the right judgment."
Pressed to elaborate, Trump said, "You look at the way he's dealt with the Senate, where he goes in there like a, you know, frankly like a little bit of a maniac. You're never going to get things done that way."
Cruz has alienated both Republican and Democratic colleagues for his tactics in the upper chamber. In pressing his staunchly conservative agenda, he has often complicated legislation pushed by leaders of both parties. In 2013, he led an effort against President Obama's signature health-care law that many said caused the subsequent government shutdown.
Trump said Cruz's unwillingness to "cajole and get along with people" makes him a flawed candidate for president.
"He'll never get anything done, and that's the problem with Ted," Trump said.
A Des Moines Register poll released Saturday showed Cruz leading the Republican race in Iowa with 31 percent of the vote. Trump placed second with 21 percent. Meanwhile, as first reported by the New York Times, Cruz voiced doubts about Trump's "judgment" in a closed-door fundraiser last week.
In an interview with CNN's "State of the Union," Trump tried to counter Cruz's remarks about his judgment by highlighting his business success.
"I would say I have far better judgment than Ted," the billionaire real estate mogul said.
Trump also faced continued criticism on Sunday for his call to bar Muslims from entering the country over concerns about terrorism.
"It does endanger national security, because it exhibits an attitude by one American who is running for the highest office of our land about a willingness to discriminate against a religion. I mean, that is against our Constitution, it is against who we are as Americans," Secretary of State John Kerry said on ABC's "This Week With George Stephanopoulos."
A Cruz spokesperson declined to comment on Trump's criticism. Cruz is seeking to play down any discord with Trump in his public remarks.