The TV presenters also said three White House officials had told them the National Enquirer would publish a negative article about them unless they apologised to Mr Trump for their scrutiny of him.
Mr Scarborough said: "They said, 'if you call the president up and you apologise for your coverage then he'll pick up the phone and basically spike the story.'"
"I had, I will just say, three people at the very top of the administration calling me."
Ms Brzezinski said reporters from the supermarket tabloid began harassing her family.
"They were calling my children," she said. "They were calling close friends.
"These calls persisted for quite some time and then Joe had the conversations that he had with the White House where they said 'oh, this could go away'."
David Pecker, chief executive of the Enquirer's parent company, is a staunch ally of Mr Trump, which is reflected in the tabloid's coverage of the president.
But in a Friday morning tweet, Mr Trump denied the couple's claims, saying Mr Scarborough "called me to stop a National Enquirer article. I said no!"
Mr Scarborough swiftly hit back: "Yet another lie. I have texts from your top aides and phone records. Also, those records show I haven't spoken with you in many months."
Mr Scarborough also said that a "well-known congressman" had told him that the president went on a "rant" against the show during a meeting with around 20 legislators at the White House to discuss efforts to pass healthcare legislation.
"I've been in politics my whole life. … He scared me," Mr Scarborough said, describing what the source had told him.
"Because he was vicious when he turned from you to Mika," he described being told.
"His face was red. He started talking about blood coming out of her ears, out of her eyes."
Mr Trump's original tweet about the MSNBC hosts provoked a storm of criticism, including even from some fellow Republicans.
Senator Lindsey Graham said Mr Trump's remarks on Thursday were "beneath the office" of president.
He said the tweet "represents what is wrong with American politics, not the greatness of America".
"Please just stop. This isn't normal," tweeted fellow Republican Senator Ben Sasse.
The Democratic leader in the House of Representatives, Nancy Pelosi, called Mr Trump's tweets "sexist, an assault on the freedom of the press & an insult to all women".
The White House sprang to Mr Trump's defence.
"I don't think that the president has ever been someone that gets attacked and doesn't push back," spokeswoman Sarah Huckabee Sanders told Fox News.
In a previous spat with journalist Megyn Kelly, Mr Trump referred to "blood coming out of her wherever", implying her work was being affected by her menstrual cycle.
The Morning Joe co-presenters have increasingly attacked President Trump since he took office.
In recent weeks, Mr Scarborough has called him a "bumbling dope", while Ms Brzezinski has called him "phony", a "narcissist" and accused him of "destroying the country".
Ms Brzezinski is the daughter of the late Zbigniew Brzezinski, who was President Jimmy Carter's national security adviser, while Mr Scarborough is a former Republican congressman.