The White House tried to limit what Fiona Hill, who until August served as President Donald Trump's top Russia analyst, could say to Congress in its impeachment inquiry, correspondence between her lawyers and a White House deputy counsel shows.

The letters, obtained by NBC News, illustrate that while the White House did not try to block Hill from testifying, it did tell Hill’s lawyers about four areas that could potentially fall under executive privilege.

Those areas involved direct communications with the president, diplomatic communications, meetings with other heads of state and staffing the president had on calls with foreign heads of state, the correspondence shows.

Hill testified under subpoena on Monday to the House. Previously, she agreed to testify at Congress' written request.

Trump impeachment inquiry: Live updates and the latest news

The House has launched an impeachment inquiry over matters that include Trump’s communication with Ukrainian President Volodymyr Zelenskiy that involved a request for an investigation by that country into former vice president Joe Biden and his son Hunter.

There is no evidence of wrongdoing by either Biden.

Former White House advisor on Russia, Fiona Hill, leaves Capitol Hill after testifying before congressional lawmakers as part of the House impeachment inquiry into President Donald Trump, on Oct. 14, 2019. Manuel Balce Ceneta / AP

Hill’s lawyers wrote back to the White House on Sunday, presenting their argument for why executive privilege did not apply, based in part on the fact that some of the information has already come into the public sphere and thus is no longer confidential, the correspondence shows.

Hill’s lawyers also made a legal argument that executive privilege disappears when there’s reason to believe there was government misconduct. Hill’s lawyers asked the White House to weigh in further, according to the correspondence.

Monday morning, via an emailed letter from White House Deputy Counsel Michael Purpura to Hill's attorney Lee Wolosky, the White House wrote back and generally dismissed most of the arguments by Hill’s lawyers.

"As the White House Counsel has explained, there is no valid impeachment inquiry underway," Purpura wrote in the letter.

The White House did not tell Hill not to testify, but reminded her in the emailed letter Monday morning that "it is incumbent on Dr. Hill and you, as her counsel, to guard against unauthorized disclosure. To be clear, Dr. Hill is not authorized to reveal or release any classified information or any information subject to executive privilege."

In her closed-door testimony, Hill planned to tell Congress that Trump's personal attorney Rudy Giuliani and U.S. Ambassador to the European Union Gordon Sondland circumvented the administration to pursue a shadow foreign policy on Ukraine, a person familiar with her expected testimony told NBC News last week.

A source familiar with Hill’s closed-door testimony Monday says it followed along the lines of NBC News' previous reporting of her planned testimony. Hill, the source said, raised concerns about Giuliani's efforts in Ukraine and also spoke in support of ousted U.S. ambassador to Ukraine Marie Yovanovitch.

A former senior White House official had told NBC News that Hill's appearance has caused concern among those close to Trump because she played a central role in the administration's Russian and Ukrainian policy.

Hill's testimony comes after Yovanovitch told House investigators last week that Trump pressured the State Department to remove her. Pushing for Yovanovitch's ouster was central to an effort from two Soviet-born business associates of Giuliani who now face federal charges over campaign-finance violations.

Sondland is scheduled to testify Thursday. He is expected to testify that he did not know why Congress- approved military aid to Ukraine was withheld, a person with knowledge of his testimony told NBC News.

Sondland is set to say he "relied" on Trump's pledge that there was no quid pro quo with regard to Ukraine investigating the Biden family and a conspiracy theory involving the 2016 presidential election when Sondland texted the current top diplomat in Ukraine last month saying no quid pro quo was taking place.