Nadler's committee had subpoenaed Barr to hand over the unredacted version of Mueller's report on Russian election meddling, possible coordination with Trump's 2016 campaign and possible obstruction of justice by Trump himself. It also asked for the underlying evidence that was used as the basis for Mueller's report.

But Nadler said in a statement Monday that, "Given our conversations with the Department, I will hold the criminal contempt process in abeyance for now."

House Democrats had threatened to hold Attorney General William Barr and former White House counsel Don McGahn in contempt for failing to comply with subpoenas. A vote was expected for Tuesday in the House.

House Judiciary Committee Chairman Jerry Nadler, D-N.Y., on Monday said he would hold off on contempt proceedings for officials in President Donald Trump's administration because the Justice Department had begun sharing special counsel Robert Mueller's "most important files" with lawmakers.

The committee subpoenaed McGahn, who was one of the most-cited witnesses in the 448-page report, to testify before the committee. The White House directed McGahn not to comply with that subpoena.

Nadler's statement did not suggest that the DOJ had agreed to hand over the entire report without any redaction, which the department has argued would violate federal rules about revealing grand jury testimony. A spokeswoman for the DOJ did not immediately respond to CNBC's request for comment on Nadler's statement.

Mueller found insufficient evidence to show coordination between the Trump campaign and Russia. He declined to conclude whether the president obstructed justice but noted that the report did not exonerate him. Barr and Deputy Attorney General Rod Rosenstein determined, based on the Mueller report, that there wasn't enough evidence to support an obstruction offense.

Read Nadler's full statement below: