"We have the votes," McConnell said, for the Senate to follow the precedent of the Clinton impeachment trial in which the question of whether to call witnesses was voted on after arguments from the prosecution and the defense.
"What's good for President Clinton is good for President Trump," he said. "We'll get around to a discussion of witnesses."
Pelosi, in a letter to colleagues Tuesday night, called on McConnell publicly to outline the trial and release the resolution the Senate would approve to begin the process.
"It is important that he immediately publish this resolution, so that, as I have said before, we can see the arena in which we will be participating, appoint managers and transmit the articles to the Senate, Pelosi wrote, echoing sentiments she shared in a closed-door meeting with members of party leadership earlier in the evening.
"She wants to know what the procedures are. Fair question," Rep. Peter Welch, D-Vt., said of Pelosi's position. "She's making the point that in order to appoint our managers, we need to know the lay of the land."
Pelosi and McConnell's back-and-forth followed a surprise statement from Bolton on Monday. The former White House aide, a key player in events at the center of the Ukraine impeachment inquiry who did not cooperate with the House investigation, announced that he would appear before the Senate to testify if subpoenaed to do so.
The news prompted new questions for some Senate Republicans thought to favor bringing new evidence into the trial. But key moderate Republican senators willing to hear from Bolton, including Utah's Mitt Romney and Maine's Susan Collins, said they favored McConnell's plans to postpone any votes to do so.
Schumer insisted that it made no sense to have senators hear arguments before all the witness and documentary evidence had been presented.
"There will be votes on the four witnesses we have asked for," Schumer said. "We Democrats have the ability to get votes in an impeachment trial."
"The eyes of America -- the eyes of the Founding Fathers looking down upon us -- are on our Republican colleagues," Schumer continued, referring to the four Republicans senators Democrats were hoping would support their position and help them form a voting majority.
If they vote against witnesses and documentary evidence, Schumer said, "they will be engaging in a cover-up. The next few weeks will be very, very telling," he said.
"We will not let them avoid the vote. They can delay it -- they can't avoid it," he said of the Republican strategy to put off the witness question.
At the White House, President Donald Trump, who repeatedly called for former Vice President and Democratic candidate Joe Biden and his son, Hunter, to testify, said he would defer to the Senate on the question of whether Bolton should testify, adding, "he would know nothing."
"That's going to be up to the lawyers. It'll be up to the Senate, and we'll see how they feel," Trump said during an Oval Office photo op. "He would know nothing about what we're talking about, because, as you know, the Ukrainian government came out with a very strong statement -- no pressure no anything and this from the boss. That's from the President of Ukraine," he said.
ABC News' Katherine Faulders contributed to this report.