(CNN) The Senate is at an "impasse" about setting the rules of the impeachment trial of President Donald Trump, Senate Majority Leader Mitch McConnell said Thursday, leaving the status of the trial in limbo as lawmakers leave Washington until the new year.

The House of Representatives took its final vote of the year Thursday without appointing impeachment managers or sending articles of impeachment to the Senate, suggesting that House Democrats won't take the steps needed to start the impeachment trial in the Senate until January.

Despite the fits and starts over the trial, Speaker Nancy Pelosi and other House Democrats are playing down the prospect of an extended standoff with the Senate over the two articles of impeachment passed by the House Wednesday, signaling that it's only a matter of time before the House does in fact transmit them to the Senate.

"We remain at an impasse on these logistics," McConnell said on the floor after a meeting with Senate Minority Leader Chuck Schumer.

"I continue to believe that the unanimous bipartisan precedent that was good enough for President (Bill) Clinton that ought to be good enough for President Trump," McConnell added, a reference to the resolution that passed 100-0 at the beginning of the Clinton impeachment trial, which did not deal with issues like witnesses.

The House's impeachment of Trump on Wednesday means that the case goes to the Senate to consider whether the President should be removed from office -- but not before the House formally transmits the articles to the Senate.

The developments Thursday as Congress left town mean that the step of sending the impeachment articles to the Senate could wait until the week of January 6, when the House reconvenes. Pelosi has said that the resolution to name impeachment managers needs to be approved before the articles could be transmitted. And she's said that she needs to understand what the Senate trial will look like before naming the House managers who will prosecute the case before the senators.

After their meeting, Schumer's spokesman said Schumer is still pushing for an agreement that includes his proposal for witnesses in the Senate trial.

"Sen. Schumer asked Sen. McConnell to consider Sen. Schumer's proposal over the holidays because Sen. Schumer and his caucus believe the witnesses and documents are essential to a fair Senate trial," Justin Goodman, Schumer's spokesman, said in a statement.

It won't take much time for the House to approve the resolution naming the managers. The Senate's first votes are expected on January 6, McConnell said, and the House's first roll-call votes are scheduled for January 7.

Pelosi told reporters Thursday that she was waiting for McConnell and Schumer to cut a deal on the rules that would govern the Senate trial. The question still looms of what happens if McConnell and Schumer don't reach a deal on the parameters.

Pelosi was careful Thursday not to say at her news conference that Democrats must get a "fair" trial before transmitting the articles. That's a different tone than Wednesday night when Pelosi suggested that they needed to have a fair trial before deciding whether to submit the articles to the Senate.

Some Democrats have suggested that Pelosi should hold the articles as leverage to win concessions from the Senate before the trial begins. But McConnell dismissed that notion Thursday.

"It's beyond me how the speaker and Democratic leader in the Senate think withholding the articles of impeachment and not sending them over gives them leverage," McConnell told reporters. "Frankly, I'm not anxious to have the trial. If she thinks her case is so weak she doesn't want to send it over, throw me into that briar patch."

McConnell added on the floor that House Democrats appeared to be getting "cold feet" to prosecute their case.

But while McConnell said it was "fine with me" if the House never sends the articles, Republican Sen. Lindsey Graham of South Carolina, a close ally of the President, said in a Fox News interview that Trump was "demanding his day in court," after meeting with him.

Trump weighed in on Twitter, saying that Pelosi "feels her phony impeachment HOAX is so pathetic she is afraid to present it to the Senate, which can set a date and put this whole SCAM into default if they refuse to show up!"

Schumer told reporters leaving a meeting in Pelosi's office that the two Democratic leaders were "on the same page."

"Speaker Pelosi and I want one thing, we want a fair trial," Schumer said earlier Thursday. "A fair trial, in my way of thinking, involves witnesses and documents. You don't have trials without them."

But when CNN asked Pelosi if there needed to be what Democrats consider a "fair" trial, she didn't explicitly say.

"Well, we'd like to see a fair process, but we'll see what they have, and we'll be ready for whatever it is," Pelosi said.

House Judiciary Chairman Jerry Nadler, a Democrat from New York, also told CNN that "I would doubt" the House holds onto the articles forever, saying they would be submitted in "due course."

McConnell has already rejected Schumer's initial request for four witnesses from earlier this week , and Republican senators have signaled they favor a quick trial without witnesses. They argue Democrats should have called those witnesses.

The two Senate leaders differ on how they see any initial agreement coming together, and that gap hasn't yet narrowed. The two sides are expected to continue to work toward an agreement over the holidays, sources said.

McConnell has left open the possibility of voting to move forward with the trial without a bipartisan agreement when the Senate returns in January.

Pelosi cited the 100-0 Senate vote that established the rules for the Clinton impeachment trial in the Senate and said she hoped a similar agreement would be struck this time.

"We would hope they would come to some conclusion like that. But in any event, we're ready. When we see what they have, we will know who and how many we will send over. That's all I'm going to say about that now," Pelosi said.

This story has been updated with additional developments Thursday.