Democratic leaders are considering dates for votes by the full House of Representatives to hold Attorney General William Barr -- and possibly others -- in contempt of Congress, two Democratic congressional aides told CBS News. House Judiciary Chairman Jerry Nadler, of New York, said there could be a vote in early June, and South Carolina Rep. Jim Clyburn, the Democratic whip, said the "target" was the week of June 4.

The House Judiciary Committee voted earlier this month to hold Barr in contempt of Congress for failing to comply with its subpoenas regarding documents related to special counsel Robert Mueller's Russia report. President Trump has asserted privilege over the entire Mueller report and underlying documents.

Earlier this week, the White House also directed former White House counsel Don McGahn not to appear before Congress, citing a newly published opinion of the Justice Department Office of Legal Counsel. He had been subpoenaed to appear before the Judiciary Committee Tuesday. Democrats had hoped to question him about what he had told the special counsel regarding Mr. Trump's orders to fire Mueller, an order that McGahn refused.

Get Breaking News Delivered to Your Inbox

Majority Whip Rep. James Clyburn, of South Carolina, told reporters Thursday morning, "We believe that the week of June 4 is our target for dealing with an issue on the floor. That was reported to us today."

House Speaker Nancy Pelosi has said she'd like to put the contempt votes on the floor together.

Reporting by Rebecca Kaplan