WASHINGTON (CN) — Senate Majority Leader Mitch McConnell said Tuesday he expects President Donald Trump’s impeachment trial to start next week, as the House nears a vote to transmit the articles of impeachment it approved last month.

McConnell said if the House votes to transmit the articles of impeachment on Wednesday as expected, preliminary matters like Chief Justice John Roberts coming to the Senate to swear in senators could take place by the end of this week. Once those steps are out of the way, the formal trial would get underway next Tuesday.

McConnell’s comment comes hours after Speaker of the House Nancy Pelosi announced she was ending a nearly month-long standoff in which she refused to send the articles of impeachment to the Senate until she saw how Republicans planned to conduct the trial.

Even with Pelosi’s standoff, that still remains somewhat unclear. McConnell announced last week that he has the votes for a preliminary plan similar to the one used in the trial for President Bill Clinton that would set out broad details of how proceedings would play out, such as the amount of time each side will have for arguments, but leave the question of witness testimony until later.

The Kentucky Republican told reporters Tuesday that his caucus’ unity has not faded on the issue and that the resolution, which has not yet been unveiled, will come up for a vote next week if the House sticks to its expected timeline.

The resolution would not resolve the issue of witnesses, which remains a live one. Democrats continue to insist the Senate cannot hold a fair trial without hearing from key people in the know about Trump’s alleged efforts to leverage a coveted White House meeting and $400 million military aid package in exchange for the Ukrainian government announcing investigations into former Vice President Joe Biden’s son and an unfounded theory that Ukraine, rather than Russia, interfered in the 2016 presidential election.

These potential witnesses include former National Security Adviser John Bolton and acting White House Chief of Staff Mick Mulvaney. Bolton has said he would be willing to testify if the Senate subpoenaed him as part of the trial.

Earlier in the impeachment process, McConnell said he plans to coordinate with the White House on how the trial will go. But he has been adamant that issues like witnesses testimony be left up to up-or-down votes in the Senate, which has a narrow Republican majority.

While Republicans have generally been opposed to hearing from witnesses as part of the trial, some have not ruled out voting to call people like Bolton. Speaking to reporters Tuesday, Senate Minority Leader Chuck Schumer expressed tepid optimism that the tide of public opinion was chipping away at the initial Republican opposition.

“If you want the truth, you have to have witnesses,” the New York Democrat told reporters Tuesday. “You have to have documents. Who has ever heard of a trial without witnesses and documents?”

But some Republicans have also said if the Senate is going to hear from witnesses during the trial, Biden’s son, Hunter, should be among them. One of the investigations Trump raised with Ukrainian President Volodymyr Zelensky on the now-infamous phone call that launched the impeachment proceedings was into Burisma, a Ukrainian energy company on whose board Hunter Biden sat.

When asked at a press conference Tuesday, McConnell did not directly endorse calling Hunter Biden as a witness at the Senate trial, but also did not rule it out. He cautioned that if there are witnesses, Democrats should not expect to have a monopoly on who the Senate calls.

“I think it’s certainly appropriate to point out that both sides would want to call witnesses that they want to hear from,” McConnell said. “So when you get to that issue, I can’t imagine that only the witnesses that our Democratic colleagues would want to call will be called.”