The committee calls McGahn the "most important witness", in its investigation into possible obstruction of justice by Trump.
In May, McGahn, now a lawyer in private practice in Washington, D.C., refused, at the direction of the White House, to comply with a congressional subpoena to appear before the House Judiciary Committee.
McGahn defied a committee subpoena to testify in May, after the White House directed him not to cooperate with the panel. The House is also seeking Trump's financial records through other federal lawsuits, as committees investigate the President's ethics and possible foreign influence over the administration.
House Speaker Nancy Pelosi referenced the lawsuit in a letter to colleagues Wednesday, in another signal that she's opening the door to considering impeachment.
"Their insistence on having Don McGahn testify publicly before the cameras further proves they are only interested in the fight and public spectacle of an investigation, but not actually in obtaining any real information", Representative Doug Collins, the panel's top Republican said in a statement.
Articles of impeachment represent a formal accusation of misconduct that would require 218 votes to pass the 435-member House.
McGahn told Mueller's investigation team that Trump pressed him repeatedly to have the special counsel removed and then to deny that he had been instructed to do so.
Invoking the Democrats' efforts to seek backing in the courts for congressional subpoenas, Cummings said, "The day that any administration disobeys a court order, then I'm for impeachment because, you see, by that time, we're in trouble". The letter notes a House subcommittee recently held a hearing on transparency in the federal courts, while lawmakers are considering legislation that would impose a code of conduct on the justices. The subpoena to McGahn is one of a number the committee has issued and been rebuffed by the White House: both former communications director Hope Hicks and former McGahn deputy Annie Donaldson did not answer almost all of the questions that the committee asked of them related to their time at the White House. The National Archives has processed records from Kavanaugh's tenure as White House counsel, according to the Nadler-Johnson letter.
But the aides argued they still needed to hear from McGahn, because he can talk about the offenses laid out in the Mueller report, as well as the President's intent in those episodes.
"Don McGahn is Donald Trump's John Dean", a second Democratic lawyer for the committee said. The request pointed to the materials from Kavanaugh's time as a staff secretary to President George W. Bush, which were not among the records requested by then-Senate Judiciary Chairman Chuck Grassley (R-IA) when Kavanaugh's nomination was before his committee.