As special counsel, Mr Mueller was tasked with investigating Russian interference in the 2016 presidential election.
Pelosi has so far resisted starting an impeachment inquiry, even as a growing number of House Democrats have said publicly they are in favour of starting one.
Criminal contempt would be referred to the justice department, where it would certainly be rejected. In the courts, meanwhile, Democrats have scored some early wins over Trump.
The House Judiciary Committee will be able to review some of the underlying evidence from the Mueller report. Pelosi, D-Calif., prefers to continue the investigations.
President Donald Trump has continued his war of words against Pelosi. After Monday's deal, however, they made a decision to vote only on the motion authorizing the panel to go to federal court, if necessary, to enforce subpoenas.
Just prior to Monday's hearing, Judiciary Committee Chairman Jerrold Nadler of NY announced that the Justice Department had agreed to provide the committee with some of the Mueller probe documents it had requested.
Cummings on Monday announced a new vote this week to recommend holding Attorney General William Barr in contempt of Congress, saying he's being stonewalled in trying to get answers about the citizenship question on the 2020 census.
Nadler's announcement comes after the Judiciary Committee voted to hold Barr in contempt last month, after the Justice Department did not comply with a subpoena seeking access to an unredacted version of Mueller's Russian Federation report, as well as underlying documents and evidence.
Nadler said in his statement he would give the justice department time to comply. "If important information is held back, then we will have no choice but to enforce our subpoena in court and consider other remedies".
The Department's Office of Legal Counsel issued an opinion last month that said congressional subpoenas for agency employee testimony are legally invalid if they require the employee to appear without an agency lawyer. Collins argues that Democrats are overstepping by pursuing lawsuits on Mueller report material. But he said McGahn is another matter.
"He's not a credible person", Conway said of Dean. "He should begin to co-operate immediately or face the consequences".
"The Mueller Report was a disaster for them", Trump wrote.
Voters were also were split on whether an investigation into how and why the original probe into Trump - which led to the appointment of Special Counsel Mueller - was started.
Implicitly urging the House Democrats to launch impeachment proceedings against Trump, he said: "In many ways the Mueller report is to President Trump what the so-called Watergate road map... was to President Richard Nixon".
In his leadoff testimony, Dean pointed to parallels between Trump's actions and the efforts of Nixon to obstruct the investigation of the Watergate special prosecutor.
"John Dean's been a loser for many years", Trump told reporters later Monday. He added his oft-repeated claim of "No Collusion - No Obstruction!"