Senate Democrats are breaking with their party leaders over House Speaker Nancy Pelosi’s decision to hold on to the articles of impeachment, saying it is time to get President Trump’s trial rolling.

The grumbling in the ranks puts pressure on Senate Minority Leader Charles E. Schumer, New York Democrat, who has stood by Mrs. Pelosi’s decision to withhold charges against the president in an attempt to get Senate Republicans to guarantee witnesses will be called during the trial.

Majority Leader Mitch McConnell used the crack in the ranks Wednesday to call on Mr. Schumer to break Mrs. Pelosi’s impasse.

Mr. Schumer did not heed the call. Instead, he continued to stress the need for more witness testimony.

“The Senate is not a creature of the House,” Mr. McConnell, Kentucky Republican, said on the chamber floor. “Stand up for the Senate. Stand up for our institutions. Stand up for the country.”

An impeachment trial in the Senate cannot begin until the upper chamber receives the charges from the House, according to precedent.

Mrs. Pelosi, California Democrat, has kept the two articles since they were voted on Dec. 18, an unprecedented move that Republicans say runs afoul of the Constitution, which dictates that the Senate conduct impeachment trials.

Sen. Doug Jones, a vulnerable 2020 Democrat from the red state of Alabama, is one of the rank-and-file lawmakers ready to get the process started.

“I’m hoping they will come over here soon. I think most people are ready to get moving on this,” he told CNN.

Still, he said it’s important for a fair trial with witness testimony to be held in the Republican-controlled Senate, echoing Mr. Schumer’s demands.

Democratic Sens. Chris Murphy and Richard Blumenthal of Connecticut, Joe Manchin III of West Virginia, and Sen. Angus King, Maine independent, have all said they are ready to have Mrs. Pelosi transmit the articles, according to multiple reports.

Sen. Dianne Feinstein of California, the top Democrat on the Senate Judiciary Committee, told reporters she does not see what good the delay does.

On the other side of the Capitol, House Democrats stood firm behind Mrs. Pelosi, including a few members from swing districts in Trump-won districts such as Rep. Andy Kim, New Jersey Democrat, who said he hasn’t heard anything from his constituents about impeachment.

“I’ve just been 100 percent focused on the Middle East crisis,” he said. “I’m not thinking about that process at all right now.”

Mrs. Pelosi was in a meeting about impeachment Tuesday night when Iran launched misiles at U.S. military bases in Iraq, shifting House Democrats’ focus.

Meanwhile, Rep. Jamie Raskin, Maryland Democrat, disagreed with the handful of his Senate colleagues who wanted to move on with the process.

“We are behind the speaker and we think she will have the best sense of when the Senate will stop their threats of nullifying the whole process,” he said.

House Democrats impeached Mr. Trump in a party-line vote Dec. 18, and after returning to the Capitol Tuesday after a two-week holiday break, Mrs. Pelosi has remained publicly mum on the subject.

The president was impeached for abuse of power and obstruction of Congress, charges that stem from Mr. Trump’s July 25 phone call with the Ukrainian president in which he requested a probe into a political rival, former Vice President Joseph R. Biden.

Mr. Schumer had wanted assurances that more witnesses and evidence would be available during the Senate trial. He expressed hope that at least four moderate Republican senators would join his request, giving Democrats the 51 votes needed to prevail in a floor vote.

He particularly wants former national security adviser John Bolton and White House chief of staff Mick Mulvaney to testify.

But Senate Republicans have remained united behind Mr. McConnell’s demand for a trial process that would mirror the 1999 impeachment trial of President Bill Clinton.

The 1999 process would leave the decision on calling additional witnesses until after House Democrats’ impeachment managers and the president’s legal team present their cases.

“What was good enough for President Clinton is good enough for President Trump,” Mr. McConnell said this week.

The GOP unity has given Mr. McConnell the upper hand in negotiations as Democrats appear to be backing off their pre-trial demands.

But House Democrats argue the Clinton impeachment precedent is different because it stemmed from a more than year-long special counsel probe.

“The entire context is so dramatically different,” Rep. Dan Kildee, Michigan Democrat, said Tuesday. “The Ken Starr investigation, all the work leading up to the impeachment of Clinton, was so dramatically different that there’s not really apples-to-apples comparison. The fact is in front of us right now we have witnesses that have information that is relevant to the determination that the Senate is going to have to make.”

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