"Lucifer in the flesh", Boehner told Stanford's David Kennedy, a history professor emeritus, according to the Stanford Daily.
Boehner said he was "texting buddies" with the other two Republican candidates; front-runner Donald Trump and Ohio Governor John Kasich, CNN reported.
The account in the student newspaper is accurate, a source close to Boehner confirmed on Thursday.
Cruz in response told reporters that Boehner "allowed his inner Trump to come out".
"The interesting thing is I've never worked with John Boehner, I don't know the man," Cruz said. "Indeed, during the government shut down, I reached out to John Boehner, to work with him to get something meaningful done. He said, 'I have no interest in talking to you.' "
Boehner and Cruz clashed over the government shutdown and Obamacare fights from the time Cruz took his Senate seat in 2013 to when Boehner stepped down late last year.
"When John Boehner calls me Lucifer, he is directing it at you. What Boehner is angry at is me standing with the American people," Cruz added.
Boehner also compared Cruz to the devil last month in a talk before financial industry lobbyists and executives in Florida.