WASHINGTON –- Two-thirds of voters want President Donald Trump’s former national security advisor, John Bolton, to testify in the Senate impeachment trial that could start next week –- a step that Texas’ two Republican senators resist.

Bolton was one of a handful of insiders with potentially crucial eyewitness testimony who did not appear before the House impeachment inquiry, because the president ordered them not to, citing executive privilege.

“If they don't think there's enough evidence to sustain those two articles of impeachment and they have to have Mr. Bolton's testimony, they sure made a big mistake in the House,” Sen. John Cornyn said last week.

A Quinnipiac University Poll released on Monday found strong public support for having Bolton testify, something he offered to do last week if the Senate issues a subpoena -- a move that ramped up pressure on Trump and his allies.

Support to hear from Bolton is strong even among Republicans.

Although only one in 20 think the president should be removed, four in 10 say they’d like to hear testimony from Bolton. As national security advisor, he likely knows firsthand why Trump froze nearly $400 in military aid for Ukraine. At the time, Trump was prodding Ukraine to publicly announce a corruption probe aimed at tarnishing Joe Biden.

Cruz insists that Bolton should only be allowed to testify if Trump’s legal team is allowed to compel testimony from Hunter Biden, the son of former vice president.

Trump has asserted that he sought the inquiry because Hunter Biden’s lucrative work with a Ukrainian energy firm, Burisma, reeked of corruption.

Majority Leader Mitch McConnell called the issue of witnesses “contentious.” Announcing Tuesday that the trial will begin next Tuesday, he insisted that senators won’t address the question until the trial is underway.

Actually, I am open to it if he has anything material to add, but as in the Clinton trial, the decision should be deferred until the parties present their case and Senators can ask questions. The Clinton/Schumer precedent should apply to President Trump, too. https://t.co/u1q0EWwGBZ — Senator John Cornyn (@JohnCornyn) January 14, 2020

Republicans control the Senate and the trial rules, with 53 of 100 votes.

Three GOP senators have indicated support for calling Bolton or other potentially damaging witnesses. But that’s not enough to avoid a deadlock and in an impeachment trial, the vice president does not get to break ties.

Utah Sen. Mitt Romney, the party’s 2016 nominee for president, said Monday that he likely will vote to hear from Bolton once the trial begins: “I presume I’ll be voting in favor of hearing from John Bolton, perhaps among others.”

Alaska Sen. Lisa Murkowski hasn’t said directly whether she would vote to compel testimony from Bolton but she did go further than most GOP colleagues in saying she can see the value of putting him under oath. “Am I curious about what Ambassador Bolton would have to say? Yes, I am,” she said recently.

The Texans are less eager.

Cornyn said recently that if Bolton does testify, whatever he has to say would likely help Trump, rather than support the case against him as Democrats expect.

Even so, Cornyn isn’t ready to support a subpoena, saying that “I would want to know what he has to offer that would help illuminate the issues.”

The subpoena question

But Bolton has refused to testify without a subpoena. Asked about that Catch-22, Cornyn blamed House Democrats for marching ahead with impeachment without waiting for courts to sort out Trump’s claims of executive privilege.

“[House intelligence chairman] Adam Schiff dropped the subpoena of Bolton like a hot potato when Bolton said he would have to go to court,” he said. “Schiff did not want to delay the proceedings.”

Trump has given mixed signals.

In the Oval Office last week, with the Greek prime minister at his side, Trump said that “He would know nothing about what we’re talking about” and in any case, Ukraine’s president “came out with a very strong statement -- no pressure, no anything.”

But Trump also indicated he would try to block Bolton’s testimony, saying days later that “we have to protect presidential privilege. When we start allowing national security advisors to just go up and say whatever they want to say, we can't do that.”

On cable news, Sunday public affairs shows and conservative talk radio, Cruz has served as a top defender of Trump, echoing complaints that impeachment is a sham, accusing Democrats of running an unfair inquiry, and glossing over the fact that Republicans got equal time for questions at House hearings and that Trump refused to let key witnesses testify, including Bolton and acting chief of staff Mick Mulvaney.

“If the prosecution gets to call John Bolton, then the president ought to get to call Hunter Biden. It ought to be fair and even and respect due process,” he said on Fox News, where he said that allowing witnesses “is going to be a hotly debated question within the Senate.”

WATCH: @SenTedCruz: "This the first time in the history of our country any president has been impeached without even so much as an allegation of criminal conduct" #nine2noon pic.twitter.com/DSz3ed9KCR — America's Newsroom (@AmericaNewsroom) January 14, 2020

On the Hugh Hewitt Show on Tuesday, each time the host invoked Bolton, Cruz sidestepped whether the former national security advisor could shed light on the core allegations.

“The Senate is likely to resolve the issue of whether witnesses are called in about two weeks, after opening arguments from House managers and the White House defense team,” Cruz said.

He said that he is “open to the possibility of the Senate hearing witnesses” but indicated the witness list should emphasize those who might exonerate Trump because the House conducted “a one-sided show trial.”

“If the president wants to call Hunter Biden, if the Senate wants to call the whistleblower, due process dictates that he have the opportunity to present his defense,” Cruz said. “If the Senate decides to allow the prosecution to call yet more witnesses – after all the witness they had in the House, they now want to call more witnesses – well I don’t know if there are going to be 51 votes to do that or not.”

How the Senate will proceed

The impeachment inquiry began after a CIA whistleblower filed a complaint airing concerns by White House and national security officials about Trump’s July 25 call with Ukraine’s president. In that call, Trump pressed his counterpart to publicly announce a corruption probe targeting the Bidens, at the time when he had frozen nearly $400 million in military aid that Ukraine needed to deter Russian aggression.

The House approved articles of impeachment on Dec. 18 accusing Trump of abusing his office by trying to coerce Ukraine to bolster his own political fortunes, then stonewall Congress by withholding documents and barring testimony from top aides.

On Fox News, Cornyn indicated that he’s open to witnesses but argued they won’t be necessary, predicting that will be clear once each side presents opening arguments because “I don’t think the facts are that disputed.”

“We will not shut the door on the possibility for additional witnesses,” Cornyn told Texas reporters last week.

He added that a decision on calling Bolton or any other witness should wait “until after the president and the impeachment managers have presented their case and senators get to ask questions. Then if there's more witnesses that 51 senators want, they can be requested as there were in the Clinton trial.”

Democrats note that when Bill Clinton was impeached, independent counsel Ken Starr had already collected volumes of evidence. Clinton himself had testified under oath, as had all other relevant witnesses. That is not the case in the Trump inquiry.

Washington correspondent Tom Benning contributed to this report.