On Thursday, a lawyer for a whistleblower told the White House that Trump's rhetoric was placing his client and family in physical danger.
Undeterred by the letter, Trump renewed his attacks on the whistleblower and lawyer on Friday, the BBC reported.
The individual's identity has so far been fiercely guarded by Democrats.
In August, the whistleblower filed a report that eventually triggered impeachment proceeding against Trump.
The report expressed concern over a phone call a month earlier in which Trump asked his Ukrainian counterpart Volodymyr Zelensky to investigate Joe Biden, a Democratic front-runner for the 2020 US presidential election.
In Thursday's letter, sent to White House counsel Pat Cipollone, the whistleblower's lawyer Andrew Bakaj cites many examples of the president's "fixation" on the identity of his client in his comments to the media, at rallies and on Twitter.
"Such statements seek to intimidate my client - and they have," Bakaj writes.
He continued: "Should any harm befall any suspected named whistleblower or their family, the blame will rest squarely with your client."
But the next day, Trump launched a fresh attack at the White House.
"The whistleblower is a disgrace to our country... and the whistleblower because of that should be revealed," he told reporters.
"And his lawyer who said the worst things possible two years ago, he should be sued, and maybe for treason."
Trump may have been referring to the whistleblower's other lawyer, Mark Zaid, who has been under fire from the president's allies over tweet posted in 2017 in which he vowed - among other things - to "get rid of him (Trump)".
Meanwhile, the President's daughter, Ivanka Trump, said in an interview with a leading media outlet that she did not believe the whistleblower's identity was "particularly relevant".
"The whistleblower shouldn't be a substantive part of the conversation," she told said, adding that the person "did not have firsthand information".
She echoed her father's view that the impeachment investigation was about "overturning the results of the 2016 election".
( With inputs from IANS )