(CNN) The House is moving toward the next phase of its impeachment inquiry , setting up a vote later this week on procedures that could quickly lead to President Donald Trump becoming the third president in US history to be impeached.

House Speaker Nancy Pelosi announced Monday the House would vote on Thursday to formalize the procedures of the impeachment inquiry into Trump and Ukraine, in what will be the first time the House will go on the record on the proceedings.

The vote signals a move into the next stage of the investigation following several weeks of closed-door depositions , as Democrats said the resolution would establish rules for public hearings, provide due process rights for the White House and allow information to be transferred to the committee that would ultimately consider the articles of impeachment.

House Democrats are discussing a time frame that would include public impeachment hearings before Thanksgiving and votes on whether to impeach Trump by Christmas, according to multiple Democratic sources. But Pelosi did not put a time frame on it at a closed-door leadership meeting on Monday to discuss the resolution and she has been hesitant to do so, as the timing is subject to change depending on how witnesses cooperate or if additional leads come up, according to multiple Democrats.

Still, the working theory among Democrats is there will be another week or two of closed depositions, and that public hearings before the House Intelligence Committee could begin as soon as the second week in November, when Congress returns from a one-week recess.

House Judiciary Chairman Jerry Nadler told CNN on Monday that it's "possible" a vote could be held in his committee on articles of impeachment before Christmas. "My preference is to do it right," the New York Democrat said, adding that the panel would have to "get reports from the other committees first" detailing their findings.

The Judiciary Committee also may hold its own public hearing before voting on articles of impeachment, but that hasn't been decided yet.

Democrats say Thursday's vote is not a formal authorization of the impeachment inquiry, but it nevertheless signals they are pushing forward with the investigation despite resistance from several witnesses inside the administration to appearing for testimony. The decision to hold a vote comes after pressure from Republicans and the White House that the chamber should do so, and it undercuts the key Trump administration talking point that the inquiry was illegitimate because it did not receive a full House vote.

At the same time, Thursday's vote could put Democrats from Republican-leaning districts in a difficult position politically: Pelosi and Democratic leaders had considered and decided against holding a formal vote to authorize the inquiry earlier this month, in part due to concerns expressed by moderates in their caucus.

Pelosi said in a letter to lawmakers Monday that the House would move forward with the vote on procedures "to eliminate any doubt as to whether the Trump administration may withhold documents, prevent witness testimony, disregard duly authorized subpoenas, or continue obstructing the House of Representatives."

"This resolution establishes the procedure for hearings that are open to the American people, authorizes the disclosure of deposition transcripts, outlines procedures to transfer evidence to the Judiciary Committee as it considers potential articles of impeachment, and sets forth due process rights for the President and his Counsel," Pelosi wrote.

Democratic sources say the resolution was necessary to set forth the exact procedures to transfer evidence from the House Intelligence Committee to the House Judiciary Committee -- and to detail the procedures for holding public hearings in the impeachment inquiry.

One key detail that will be voted on this week: House Intelligence Chairman Adam Schiff wants to allow staff members to also participate in the questioning, according to multiple sources.

Multiple sources said the resolution this week surprised even top members and those in leadership who were unaware of these plans.

It is not officially a vote to authorize the inquiry, the Democratic sources say, although they will argue that the White House will have no grounds to resist their subpoenas after this vote establishes the procedures as they head into the next steps.

Still, Pelosi had previously said that she didn't need to hold a vote on the impeachment inquiry at all, calling it a "Republican talking point."

"There's no requirement that we have a vote so at this time we will not be having a vote and I'm very pleased with the thoughtfulness of our caucus with the path that we are on," Pelosi said on October 15. "We're not here to call bluffs. We're here to find the truth, to uphold the Constitution of the United States."

Republicans have attacked impeachment process

But the vote will help Democrats push back on the Republican argument that Democrats are running an illegitimate impeachment inquiry behind closed doors. GOP lawmakers have repeatedly attacked Democrats for failing to vote on the inquiry, while the White House has argued it does not need to cooperate as a because the investigation is illegitimate.

The White House and House Republicans criticized Democrats' plan to hold a vote, continuing their attack on the process of the impeachment investigation.

"We won't be able to comment fully until we see the actual text, but Speaker Pelosi is finally admitting what the rest of America already knew -- that Democrats were conducting an unauthorized impeachment proceeding, refusing to give the President due process, and their secret, shady, closed door depositions are completely and irreversibly illegitimate," White House press secretary Stephanie Grisham said in a statement.

But aides inside the West Wing are still privately figuring out what their strategy should be.

Until now, they have argued that Democrats were conducting an illegitimate investigation in secret — two aspects that could change with the vote to formalize it and hold public hearings.

The thinking inside the White House for the last several weeks has been that if Democrats vote to authorize an inquiry, the administration will no longer be able to argue that Democrats have no legislative purpose to their investigations. It would likely rob Republicans of the talking point that this is a sham investigation that has denied Trump his due process rights. It could force Republicans to solely focus on defending the President's actions, instead of attacking the process, which so far several key figures of his party have been hesitant to do.

But there could be some benefits for the White House. A formal floor vote could mean that the White House is able to review the evidence that is gathered, have counsel present and cross-examine witnesses during depositions on Capitol Hill.

So far, Trump's aides have been forced to learn about the developments from news reports, and are not given transcripts of what's said behind closed doors. Right now, officials say they are waiting to see what the text of the resolution says — though it's unclear where they'll go from there.

Trump's allies in Congress are still attacking the process.

House Minority Leader Kevin McCarthy tweeted Monday that "it's been 34 days since Nancy Pelosi unilaterally declared her impeachment inquiry."

"Today's backtracking is an admission that this process has been botched from the start," McCarthy, a Republican from California, wrote in his tweet. "We will not legitimize the Schiff/Pelosi sham impeachment."

Rep. Mark Meadows, a North Carolina Republican, said that the vote was "not the same thing" Republicans have been pushing for unless it was to formally authorize the inquiry.

"House Democrats now suddenly saying they'll vote on an impeachment resolution to 'ensure transparency' is rich—considering they've spent weeks conducting interviews in secret, leaking their own talking points while locking down any and all information that benefits the President," Meadows tweeted.

The Republican argument hadn't stopped many Trump administration officials from testifying under subpoena, but on Monday former National Security Council official Charles Kupperman defied a subpoena after he filed a lawsuit Friday asking the courts to intervene and decide whether he had to comply with the subpoena.

"Given the issue of separation of powers in this matter, it would be reasonable and appropriate to expect that all parties would want judicial clarity," Kupperman said in a statement Monday.

Courts back House inquiry

Still, Democrats appeared to be on solid legal footing without a vote -- a federal judge told the House as recently as Friday that a vote authorizing the impeachment inquiry wasn't needed.

Chief Judge Beryl Howell of the DC District Court devoted six pages exclusively to this idea in a ruling that the House should have access to former special counsel Robert Mueller's grand jury material, writing that "no House 'impeachment inquiry' resolution is required" for Congress to receive information it's seeking.

"Even in cases of presidential impeachment, a House resolution has never, in fact, been required to begin an impeachment inquiry," the judge wrote. "The reality is that DOJ and the White House have been openly stonewalling the House's efforts to get information by subpoena and by agreement."

The Justice Department on Monday appealed that ruling.

It's not clear whether Thursday's floor vote signals that the Democrats are ready to move into the public phase of the impeachment inquiry, although public hearings could begin by the middle of November.

Schiff declined to say whether the resolution meant closed-door depositions would be ending soon. The three committees leading the impeachment investigation have depositions scheduled through this week, and three more officials who have been subpoenaed are slated to appear next week, although it's unclear whether any of them will do so.

"We still have further depositions to do, and I don't want to be committed to a particular timetable, but we are moving with all expedition," Schiff told reporters.

The House impeachment resolution is being introduced by the House Rules Committee, which will mark up the measure on Wednesday before the Thursday floor vote.

"This is the right thing to do for the institution and the American people," House Rules Chairman Jim McGovern, a Massachusetts Democrat, said in a statement.

This story has been updated with additional developments Monday.