(Updates with Cheney supporting impeachment, Republican efforts on vote)
By Richard Cowan and Susan Cornwell
WASHINGTON, Jan 12 (Reuters) - President Donald Trump's iron grip on his party showed further signs of weakening on Tuesday as at least three Republicans, including a member of the House leadership, said they would vote to impeach him after his supporters stormed the Capitol.
Liz Cheney, the No. 3 Republican in the House of Representatives, said: "There has never been a greater betrayal by a President of the United States of his office and his oath to the Constitution," as the Democratic-led chamber moved forward on a path to remove Trump from office.
Trump "summoned this mob, assembled the mob, and lit the flame of this attack" on the Capitol last Wednesday, Cheney, the daughter of former Republican Vice President Dick Cheney, said in a statement, adding: "I will vote to impeach the president."
Two other Republican House members, John Katko and Adam Kinzinger, said they would also vote for the historic second impeachment of the Republican president, who leaves office in just eight days.
Their announcements came as Republican leaders in the U.S. House of Representatives on Tuesday refrained from urging their members to vote against impeaching Trump, saying it was a matter of individual conscience after his supporters ransacked the Capitol.
The House plans to vote as soon as Wednesday on an article of impeachment charging Trump with inciting insurrection unless he resigns or Vice President Mike Pence moves to oust him under a provision in the U.S. Constitution.
The House was set to vote on Tuesday on a separate non-binding resolution that would call on Pence to utilize the 25th Amendment of the U.S. Constitution to remove Trump, a procedure that has never been attempted in U.S. history.
The New York Times reported that the Republican majority leader of the U.S. Senate, Mitch McConnell, was said to be pleased about the Democratic impeachment push, suggesting Trump's party was looking to move on from him after last week's stunning attack on Congress.
McConnell believes the impeachment effort will make it easier to purge Trump from the party, the Times said.
Making his first public appearance since last Wednesday's riot, Trump showed no contrition for remarks he made to supporters at a rally before they stormed the seat of Congress and also lambasted Democrats for pushing ahead with a drive to impeach him for an unprecedented second time.
"What I said was totally appropriate," Trump told reporters as he left for a trip to the U.S.-Mexico border wall near Alamo, Texas, his first public foray since the assault on the Capitol. "I want no violence."
In a debate ahead of the House vote on the 25th Amendment resolution, Democrats pushed Republican lawmakers to disavow Trump's false allegation that Democratic President-elect Joe Biden's victory in the Nov. 3 election was illegitimate - the claim that enraged Trump's supporters and prompted the violence in Washington that killed five including a police officer.
Republicans refused to concede the point and said their unsuccessful effort last week to challenge the results of the election was justified.
The resolution calls on Pence to invoke Section 4 of the 25th Amendment, a never-before used power that allows a majority of the Cabinet to strip the president of power if he or she is deemed unable to discharge the office's duties.
Pence advisers say he is opposed to the idea, and Trump told reporters in Texas he was not worried about the prospect.
With only eight days left in Trump's term, chances the Democratic push will result in his removal before Biden takes office on Jan. 20 appear remote. But Democrats say Trump's actions demand a response.
"Our nation, our democracy and our freedom cannot risk another day of the Trump presidency," said Representative Jim McGovern, the chairman of the House Rules Committee.
Democrats could also use an impeachment trial to push through a vote blocking Trump from running for office again.
If Trump is impeached by the House, he would have a trial in the Senate to determine his guilt. A two-thirds majority of the Senate is needed to convict him, meaning at least 17 Republicans in the 50-50 chamber would have to vote for conviction.
But only a simple majority is needed to disqualify Trump from future office. There is disagreement among legal experts as to whether conviction is necessary for disqualification. A different part of the Constitution, the 14th Amendment, also provides a procedure for disqualifying Trump from future office with a simple majority of both chambers.
Trump has said he plans to run again in 2024.
At a Rules Committee session setting the timeline and procedures for Tuesday's debate, lawmakers previewed the potentially emotional battle over the resolution with angry exchanges over Republican efforts to cast doubt on Biden's sweeping election win.
McGovern challenged Republican Representative Jim Jordan, a staunch Trump ally who received a Presidential Medal of Freedom from the president on Monday, to declare that Biden won "fair and square and the election was not stolen."
Jordan refused to utter those words and defended his challenges to the Electoral College result, saying: "I followed the process the Constitution prescribes" when there are concerns over a state's election results.
"I am stunned that after all that has happened we cannot get a definitive answer," McGovern responded as the two lawmakers sparred and talked over each other.
If Trump has not stepped down and Pence has not taken action by Wednesday, Democratic leaders plan to bring impeachment to the House floor.
McConnell has said no trial could begin until the chamber returns from its recess on Jan. 19.
But Democratic Minority Leader Chuck Schumer, who is set to become the majority leader after two Democrats from Georgia are seated and Vice President-elect Kamala Harris is sworn in, told reporters the Senate could be recalled to handle the matter.
(Reporting by Richard Cowan and Andrea Shalal; Additional reporting by Susan Cornwell, David Morgan, Doina Chiacu and Steve Holland; Writing by John Whitesides and James Oliphant; Editing by Scott Malone, Paul Simao and Peter Cooney)
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