WASHINGTON — As a mob was attacking the Capitol chanting “Hang Mike Pence,” a lawyer who plotted with President Donald J. Trump and his allies to try to overturn the 2020 election sent a hostile message to the vice president’s top lawyer, blaming Mr. Pence for the violence.

The email, reported by The Washington Post and confirmed by a person briefed on its contents, shows the extent to which Mr. Trump and his advisers sought to pressure the vice president, who was presiding over the certification of the election, to defy the will of the voters and keep Mr. Trump in office.

“The ‘siege’ is because YOU and your boss did not do what was necessary to allow this to be aired in a public way so that the American people can see for themselves what happened,” the lawyer, John Eastman, wrote to Greg Jacob, Mr. Pence’s chief counsel.

Mr. Pence and Mr. Jacob were in a secure room at the Capitol when Mr. Eastman sent the message, The Post said. Outside, rioters were storming the building where Congress was meeting to certify Joseph R. Biden Jr.’s election, erecting a gallows, hunting for Speaker Nancy Pelosi and forcing lawmakers to evacuate in a scene of violence and mayhem. Police officers recovered guns, knives, Tasers, Molotov cocktails, explosive devices and zip ties in the area.

Mr. Eastman rose within Mr. Trump’s legal circle in 2020 from a little-known conservative academic to one of the most influential voices in the president’s ear, writing a memo laying out steps he argued Mr. Pence could take to keep Mr. Trump in power — measures Democrats and anti-Trump Republicans have likened to a blueprint for a coup.

The two-page memo written by Mr. Eastman and circulated to the White House in the days before the certification was revealed in the book “Peril” by the Washington Post reporters Bob Woodward and Robert Costa. In the memo, Mr. Eastman argued that as vice president, Mr. Pence was “the ultimate arbiter” of the election, essentially saying he had the power to determine who won, and that “we should take all of our actions with that in mind.”

That view was rejected by Mr. Pence and his lawyers as illegal and unethical.

Since his memo was made public, Mr. Eastman has sought to distance himself from its contents, arguing that the document was taken out of context. He told National Review that the strategy was not “viable” and would have been “crazy” to pursue.

But the email to Mr. Jacob undercuts those comments. Mr. Jacob wrote in a draft opinion article that Mr. Eastman showed “a shocking lack of awareness” as “a band of thugs who had been sold the lie that the vice president had the power to reverse the outcome of the election” stormed the Capitol.

“Vice President Pence rejected the spurious legal theories that were pitched to him, and he did his duty,” Mr. Jacob wrote. “An inquiry should be made into whether the president’s outside lawyers did theirs.”

Through a spokesman, Mr. Eastman declined to comment.

The liberal activist Lauren Windsor recently recorded Mr. Eastman calling Mr. Pence an “establishment guy” and standing by the contents of his memo. “These guys are spineless,” he said of lawmakers who refused to join his plan to overturn the election based on false claims of widespread voter fraud.

In the Oval Office the day before Congress’s electoral vote certification, Mr. Trump pushed Mr. Pence to join the plot in a string of encounters, including one meeting that lasted at least an hour. Mr. Eastman was in the office and argued to Mr. Pence that he had the power to act.

Just before Mr. Pence headed to the Capitol on Jan. 6, Mr. Trump called the vice president’s residence to push one last time.

“You can either go down in history as a patriot,” Mr. Trump told him, according to two people briefed on the conversation, “or you can go down in history as a pussy.”

Mr. Eastman spoke at the rally that preceded the Capitol assault and pushed baseless claims of widespread voter fraud. As outrage grew in the days after the riot, he abruptly retired from his job as an endowed professor and constitutional law scholar at Chapman University.

The House committee investigating the attack has demanded that the White House turn over documents and communications concerning Mr. Eastman. Investigators also plan to ask him, potentially under subpoena, to sit for an interview as they scrutinize the circumstances surrounding the largest assault on the Capitol since the War of 1812. About 140 officers were injured in the violence, which claimed several lives.

Maggie Haberman and Michael S. Schmidt contributed reporting.