Giuliani had no immediate comment.
Oversight Chairman Elijah Cummings, D-Md., and Intelligence Committee Chairman Adam Schiff, D-Calif., said Cohen's concern for safety was "legitimate" given the "attacks" by Trump and Giuliani.
But they also said they expected Cohen to appear before their committee at some point.
Trump last week urged people to "watch" Cohen's father-in-law, Fima Shusterman, who was placed on probation in the mid-1990s after pleading guilty in a case in which he was charged with conspiring to defraud the IRS.
And Trump's other comments about Cohen days before that spurred Cummings, Schiff and House Judiciary Committee Chairman Jerrold Nadler, D-N.Y., to warn the president against trying to "discourage intimidate or otherwise pressure a witness not to provide testimony to Congress."
Giuliani, in an appearance on CNN last Sunday, had said Cohen is lying about Trump, and also said that Shusterman may have ties to organized crime.
A former New York City mayor and federal prosecutor, Giuliani, further said that Shusterman was involved in criminal activity with Cohen, and that Cohen is withholding information about that activity because testifying about it would be dangerous for his father-in-law.
A source close to Cohen told NBC News that Cohen's wife and father-in-law were particularly afraid about Cohen's upcoming scheduled appearance in Congress, and that they feel directly targeted by Trump.
Cohen's wife, Laura, was afraid of going to the hearing in person because she believed she was at risk of being attacked.
"The threats are real," the source says, "Trump knows what he's doing."
Cohen had served for years as Trump's personal lawyer, and once bragged of being willing to take "a bullet" for the president.
But he fell out with Trump in early 2018, after FBI agents raided his office and several residences in New York as part of a criminal investigation.
Cohen, 52, pleaded guilty later in the year to financial crimes, campaign finance violations and to lying to Congress.
Cohen admitted facilitating hush money payments to two women, porn star Stormy Daniels and Playboy model Karen McDougal, shortly before the 2016 presidential election to keep them quiet about their alleged affairs with Trump. The president has denied having the affairs.
Cohen also confessed to misleading Congress in 2017 about when an aborted plan to build a Trump Tower in Moscow actually ended, and about the extent of Trump's involvement in that project.
For months before he was sentenced in December, Cohen had cooperated with special counsel Robert Mueller, prosecutors in New York and other authorities who are investigating Trump.
His continued cooperation with Mueller after his sentencing was expected to lead to limitations on the information he could share during his testimony before Congress.
On Jan. 12, Trump, during an interview with Jeanine Pirro on Fox News, blasted Cohen, as he has done on prior occasions.
"Look, I was a client of his," Trump said. "You're supposed to have lawyer-client privilege, but it doesn't matter because if I'm a very honest person, frankly."
"But he's on trouble on some loans and fraud and taxi cabs and stuff that I know nothing about. And in order to get his sentence reduced, he says, 'I have an idea, I'll give you some information on the president.'"
Trump then told Pirro that Cohen "should give information maybe on his father-in-law, because that's the one that people want to look at. Because where does that money — that's the money in the family."
"And I guess he didn't want to talk about his father-in-law – he's trying to get his sentence reduced. So it's pretty sad. It's weak and it's very sad to watch a thing like that," the president said.
On Wednesday, Cummings and Schiff, the chairmen of the House committees, reiterated a warning to Trump that they made on this heels of his comments to Pirro.
"As we stated previously with our colleague, Chairman Jerry Nadler of the Judiciary Committee, efforts to intimidate witnesses, scare their family members, or prevent them from testifying before Congress are textbook mob tactics that we condemn in the strongest terms," Cummings and Schiff said.
"Our nation's laws prohibit efforts to discourage, intimidate, or otherwise pressure a witness not to provide testimony to Congress. The President should make no statement or take any action to obstruct Congress' independent oversight and investigative efforts, including by seeking to discourage any witness from testifying in response to a duly authorized request from Congress."
"We understand that Mr. Cohen's wife and other family members fear for their safety after these attacks, and we have repeatedly offered our assistance to work with law enforcement to enhance security measures for Mr. Cohen and his family," Cummings and Schiff said in a joint statement.
However, the chairmen, in their joint statement, also said that when their committees began talking to Cohen about appearing before them, "not appearing before Congress was never an option."
"We will not let the President's tactics prevent Congress from fulfilling our constitutionally mandated oversight responsibilities. This will not stop us from getting to the truth," Cummings and Schiff said.
"We expect Mr. Cohen to appear before both Committees, and we remain engaged with his counsel about his upcoming appearances."