NEW YORK, NY - NOVEMBER 15: Don McGahn, general counsel for the Trump transition team, gets into an elevator in the lobby at Trump Tower, November 15, 2016 in New York City. President-elect Donald Trump is in the process of choosing his presidential cabinet as he transitions from a candidate to the president-elect. (Photo by Drew Angerer/Getty Images)
Drew Angerer/Getty Images North America/Getty Images
NEW YORK, NY - NOVEMBER 15: Don McGahn, general counsel for the Trump transition team, gets into an elevator in the lobby at Trump Tower, November 15, 2016 in New York City. President-elect Donald Trump is in the process of choosing his presidential cabinet as he transitions from a candidate to the president-elect. (Photo by Drew Angerer/Getty Images)
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(CNN) —  

The White House may seek to prevent former White House counsel Don McGahn from complying with a subpoena issued by the House Judiciary Committee that demands his cooperation in its probe into potential obstruction of justice, CNN has learned.

A source familiar with the discussion said the White House could assert executive privilege over specific aspects if McGahn testifies, but noted that discussions are not that far along yet and added that other measures are still being considered.

Sources familiar with the matter said that the House Judiciary Committee has had discussions with McGahn’s attorney, William Burck, but has not been told yet whether McGahn will comply with the subpoena, which requests documents by May 7 and his testimony by May 21. Burck did not respond to a request for comment.

House Judiciary Chairman Jerry Nadler issued a statement Tuesday standing by the committee’s subpoena.

“The Committee has served a valid subpoena to Mr. McGahn. We have asked him to supply documents to the Committee by May 7 and to testify here on May 21. Our request covers the subjects described by Mr. McGahn to the Special Counsel, and described by Special Counsel Mueller to the American public in his report,” Nadler said.

“As such, the moment for the White House to assert some privilege to prevent this testimony from being heard has long since passed. I suspect that President Trump and his attorneys know this to be true as a matter of law – and that this evening’s reports, if accurate, represent one more act of obstruction by an Administration desperate to prevent the public from talking about the President’s behavior,” Nadler said. “The Committee’s subpoena stands. I look forward to Mr. McGahn’s testimony.”

Before the subpoena was issued, McGahn’s attorney sent a letter to the House Judiciary Committee stating that they need to ask the White House for documents, not his client, according to a person familiar with the letter. He also told lawmakers that any documents related to the campaign would have to come from the campaign lawyers, this person said.

This comes as the White House is escalating its efforts to push back against congressional inquiries that they argue cross the line.

On Monday, the White House instructed former security director Carl Kline not to show up for a scheduled deposition on security clearances because House Oversight Chairman Elijah Cummings refused to allow a White House attorney to be present. And the White House will likely instruct John Gore, a top Justice Department official, not to show up for scheduled testimony if he cannot have a Justice Department lawyer present, an official told CNN. The committee earlier this month issued a subpoena for his testimony.

Democrats want to question Gore about the decision to add a citizenship question to the 2020 Census. Gore met with the committee back in March, but Cummings said he refused to answer multiple questions by citing ongoing litigation.