The Senate’s top Democrat on Sunday laid out his vision for an impeachment trial of President Trump, saying the chamber should demand testimony from four witnesses who did not testify to the House’s probe.

Sen. Charles E. Schumer said the Senate should have a “fair and open” trial, saying the Senate should follow the bipartisan rules laid out during the 1999 trial of President Clinton.

“The trial must be one that not only hears all of the evidence and adjudicates the case fairly; it must also pass the fairness test with the American people,” he said in a letter to Senate Majority Leader Mitch McConnell, the top Republican. “That is the great challenge for the Senate in the coming weeks.”

If the Senate does follow that precedent it will be more than the House, which rejected previous rules and struck out on a far more partisan path toward impeachment.

Mr. Schumer called for a speedy start to the trial, with rules adopted on Jan. 6, senators sworn in as jurors the next day, and the House be called to make its case for impeachment Jan. 9.

He said the House should be given 24 hours, and the president’s team would get 24 hours to make its rebuttal.

He also shot down the suggestion from some Republicans that the allegations against Mr. Trump are so outlandish that there’s no reason to hear witnesses, and the case can be heard based on the presentations from the House and the president.

“In the trial of President Clinton, the House Managers were permitted to call witnesses, and it is clear that the Senate should hear testimony of witnesses in this trial as well,” Mr. Schumer wrote.

Ironically, Mr. Schumer voted against that same proposal in the 1999 trial of Mr. Clinton.

At the time, Democrats complaint that calling witnesses was political theater.

Mr. Schumer’s letter is not likely to sit well with Senate Republicans.

Mr. McConnell had previously said he wanted to sit down with Mr. Schumer to talk personally about the upcoming trial. His office said that’s still their plan.

“Leader McConnell has made it clear he plans to meet with Leader Schumer to discuss the contours of a trial soon. That timeline has not changed,” said McConnell spokesman Doug Andres.

Mr. Schumer said if he gets his way and witnesses are allowed, the Senate should hear from acting White House Chief of Staff Director Mick Mulvaney, his senior advisor Robert Blair, former National Security Advisor John Bolton, and Michael Duffey, the associate director for national security at the White House budget office.

All were called to testify in the House but none appeared.

Left off Mr. Schumer’s list is former Vice President Joseph R. Biden, whom House Republicans had tried to force to testify but whom Democrats refused to call. Likewise Mr. Schumer did not mention the whistleblower who kicked off the impeachment push with a complaint about Mr. Trump’s July phone call with Ukraine’s president, where Mr. Trump asked that country to open an investigation into Mr. Biden.

House Democrats are expected to vote on two articles of impeachment this week stemming from that phone call and the subsequent House investigation.

The first article accuses Mr. Trump of abuse of power as the president, saying he has invited foreign meddling in the 2020 election.

The second article accuses the president of obstruction of Congress because he directed witnesses not to testify and he shielded documents from the investigation, in both cases citing presidential immunity from congressional demands.

Republicans argue that Democrats short-circuited the usual rules of fairness and said Mr. Trump has raised important questions of privilege that the courts should be allowed to sort out before any impeachment push.

House Democrats’ impeachment clock is running far quicker than the Clinton proceedings — a fact that Republicans cite as evidence their process has not been fair.

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