(CNN) A day after a White House official suggested an impeachment trial of President Donald Trump should include live witnesses in the chamber, a senior GOP senator warned such an approach could lead to a "three-ring circus."

"That's an option," said a skeptical Republican Sen. John Cornyn of Texas about the White House idea, making clear the Senate would ultimately have to decide how a trial would be conducted, regardless of what the administration might desire.

"I don't want a three-ring circus," said Cornyn, a former Texas state judge. "It's always unpredictable and I think one alternative would be to do depositions and then present excerpts."

That is how witness testimony was handled 20 years ago during the impeachment trial of President Bill Clinton.

On Wednesday, after White House Counsel Pat Cipollone and other officials met in the Capitol with Senate Republicans to discuss expectations for a trial, Trump legislative affairs director Eric Ueland told reporters the White House wants to put on a "full defense" of Trump and would want witnesses to appear.

"We believe very strongly — given the fatally flawed process in the House — that if they were to elect against our better advice, (and) send over impeachment to the Senate, that we need witnesses as part of our trial and a full defense of the President on the facts," he said.

Ueland said the White House wants witnesses to testify in person, on the Senate floor -- something senators involved in the 1999 trial of Clinton took pains to avoid out of concern for how it would make the institution look.

Sen. John Thune of South Dakota, the Republican whip, told reporters that the entire Senate trial is "very fluid. It's hard to handicap it."

But he gave one sneak peek: "I would be hard pressed to think there would be a high level of interest in dragging this out."

Thune cautioned that discussions about witnesses have come from individual senators and no decisions have been made about who to call.

The issue is complicated because some Republicans, for instance, want to call Hunter Biden to advance GOP claims that his work for an Ukrainian energy company was corrupt. Democrats believe the allegations against Biden are several degrees of separation removed from the central charges against Trump.

While Republicans who control the chamber could force testimony on a majority vote, there's no assurance all Republicans would go along with such a plan.

Cornyn was asked about House Speaker Nancy Pelosi's announcement Thursday to move forward with articles of impeachment , Cornyn said it was "disappointing" and would have a chilling effect on legislation already stalled by partisanship.

"It's very disappointing. It threatens the passage of the (United States-Mexico-Canada Agreement), the trade deal, the highway bill," he said. "The prescription drug price bill we could be working on. All of that has been put on the shelf in favor of this partisan impeachment."

"There just doesn't seem to be a whole lot of bipartisan cooperation right now," he complained. "I think one of the reason why the USMCA is bogging down is because Speaker Pelosi doesn't want the President to get a win."

Cornyn blasted Pelosi's overall handling of impeachment.

"I think she's lost control," he said. "I mean in March she said there wasn't going to be impeachment unless it was bipartisan and now she abandoned that."

Cornyn repeated what is now a GOP talking point that the allegations against the President don't rise to the level of impeachment.

"They are trying to put a square peg into a round hole when it comes to impeachment in order to try to justify something that I don't think is impeachable."

He said discussions have not started between Senate Majority Leader Mitch McConnell and Senate Minority Leader Chuck Schumer to set the "rules of the road" for a trial, something that might determine whether witnesses will appear in the chamber or via videotaped deposition.

"I think everyone has an interest in getting this done and getting it behind us," he said about finding a path to an efficient trial process, pointing to Democratic presidential candidates he assumes would rather be campaigning than sitting in the Senate jury box.

Sen. Lindsey Graham, a Republican from South Carolina who is an ally of the President, said he doesn't need to hear from any witnesses -- live or on tape.

"I don't see how witnesses are going to help me make my mind up," he said. "Everything is as clear as a bell to me."

But Louisiana Republican Sen. John Kennedy said he think it's important that witnesses be heard and cross-examined.

"I believe in allowing both sides to have full and unfettered access to address their case. Now, if someone wants to take a year, that's a different story," Kennedy said. "One of my disappointments with the House proceedings is that they are rigged. It's just not fair when one side can't call witnesses, can't cross examine, can't offer rebuttal evidence."

Sen. Ted Cruz, a Republican from Texas, agreed.

"If the White House defense team wants to call witnesses, if they want to call Hunter Biden or the whistleblower, two natural fact witnesses that are on their face relevant to the issues before the body, I think it's only appropriate they be allowed to present their case," Cruz said. "I think the House managers likewise should be able to present their case."

Asked if he thinks at least 51 senator would vote to call those witness, Cruz said, "I don't know. But I do think there is widespread sentiment that the Senate wants to conduct a fair proceedings. What's happening in the House, sadly, is more of a show trial."