(CNN) President Donald Trump asked Attorney General William Barr to hold a news conference clearing him of legal wrongdoing with regard to his phone call pressuring the Ukrainian President to investigate Democrats, a person familiar with the matter tells CNN.

Trump has raised the idea in conversations surrounding the ongoing impeachment inquiry over recent weeks, and has said he thought the idea could help project the message that he hadn't done anything wrong, the person familiar with the matter said.

The source couldn't say whether Barr and Trump had formally discussed the idea or whether Barr had ruled it out. But The Washington Post, which first reported the story, said the attorney general refused.

Citing people familiar with the matter, the paper reported that Trump's request came sometime around the White House's September 25 release of a transcript of the call. It was conveyed to White House officials and then the Justice Department. Trump has brought up Barr's refusal to aides over the past few weeks and how he wishes the attorney general had held the news conference, Trump advisers told the Post.

In a series of Twitter posts Thursday, Trump repeatedly denied the Post's report and attacked the media, at one point saying he never asked Barr to publicly defend his phone call. However, the Post's story doesn't say Trump asked Barr directly.

Even without a news conference from the attorney general, the Justice Department gave the President nearly everything he wanted. In an orchestrated rollout alongside the release of a transcript of Trump's Ukraine call, the department publicly announced that criminal division prosecutors had found no wrongdoing by the President, at least as it relates to campaign finance law.

The department also released a legal memo on why the intelligence community's inspector general was not required to turn over a whistleblower complaint to Congress.

Despite efforts to publicly declare the matter as case closed, it is clear that the Ukraine interactions and the role of Rudy Giuliani, the President's personal lawyer, are still at the heart of an investigation led by federal prosecutors in New York.

Barr and Trump continue to have a good relationship, people close to the administration told the Post, while acknowledging that the department has at times positioned itself against the White House of late.

People close to Barr told the paper that the attorney general, though a backer of presidential power, has acknowledged the possibility of having to assert his department's autonomy.

"My take is that Barr hasn't changed one bit, that he has had a healthy distance from the beginning," a person close to the administration told the Post. "He knows the parameters of the relationship between a President and an AG."

This story has been updated to reflect additional reporting.