The move comes ahead of a full House of Representatives vote expected to be scheduled for Tuesday. The Rules Committee is expected to mark up the resolution on the day before the full vote.
House Rules Chairman James McGovern, Massachusetts Democrat, officially introduced the measure.
Democrats are seeking to hold both men in contempt for refusing to comply with congressional subpoenas issued by House Judiciary Chairman Jerold Nadler, New York Democrat.
Mr. Barr has refused to turn over a full, unredacted version of special counsel Robert Mueller’s report, while Mr. McGahn dodged a subpoena to appear and turn over documents to the Judiciary Committee.
In a letter Thursday to House Oversight Committee Chairman Elijah Cummings, Maryland Democrat, Assistant Attorney General Stephen Boyd said the Justice Department is still willing to work out the dispute and possibly turn over more documents.
“Given the ongoing accommodation efforts by the Department, we believe that a vote on contempt would be entirely premature,” Mr. Boyd wrote. “We hope that the committee will not take such an abrupt measure to terminate the accommodation process, but will instead work in good faith to respect the legitimate prerogatives of our co-equal branches.”
The resolution would give any committee chair the authority to ask a federal court to enforce subpoenas as long as the move is approved by the Bipartisan Legal Advisory Board.
Although these are the first two contempt votes, Democrats could potentially add more Trump officials.
In the case of Mr. Barr, he was subpoenaed for the full Mueller report along with the supporting evidence shortly after its April release. The Justice Department has resisted the subpoena, saying the redactions include confidential grand jury information, which cannot be disclosed under federal law.
Mr. Barr has offered certain lawmakers the opportunity to review a less redacted report, but Democrats have rejected the overture, saying they want the whole thing.
Mr. McGahn, meanwhile, did not show up for a Judiciary Committee hearing on May 21 after the White House ordered him not to testify. The administration maintains that former government officers are immune from congressional testimony.
• Gabriella Munoz contributed to this report.
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