Appearing on "CBS This Morning" on Monday, Trump was dismissive of the contentious exchanges with Bush during the debate, and of Bush himself.
"You look at Jeb -- Jeb's a loser," Trump said. "He's spent $110 million so far, he's at the bottom of the pack or toward the bottom of the pack. He's going nowhere. And the only thing he does is attack me."
Trump and Bush have had a running war of words for most of the time Trump has been running for president. The two sharply attacked each other during Saturday's debate, and Bush called Trump a loser on the campaign trail Sunday.
Trump responded Monday by calling Bush sad.
"He thinks if he attacks me that's going to show he's tough, but every time he attacks me he melts like butter."
Bush fired back on twitter later Monday morning, writing: "@realDonaldTrump, you aren't just a loser, you are a liar and a whiner. John McCain is a hero. Over and out."
Bush was referencing the time Trump caused a controversy when he questioned whether Sen. John McCain -- who spent five years as a prisoner during the Vietnam War -- is a war hero.
During Trump's interview on CBS This Morning, the incident came up when he was asked to respond to a video of Bush attacking him over his remarks about McCain, among other comments, and Trump accused Bush of mischaracterizing what he said.
Later in the interview, Trump also turned his attention to Marco Rubio, the Florida senator running second in New Hampshire, according to some recent polls. Rubio was criticized after repeating the same lines throughout the evening during a GOP debate over the weekend.
"I was standing next to him, he was to my right, and he was having a hard time," Trump laughed. "He made the one statement -- and I'm one with a good memory -- so you know, I'm hearing the one statement and then a couple minutes later he made the same statement, almost identical."
Trump added later, "You can use a soundbite once but you can't use it five times."