Eighty-seven percent of the registered voters surveyed between June 10-16 had at least heard of the allegations against Swallow, who took office in January. Almost that many, 78 percent, said he should step down.
Nearly three-quarters 71.5 percent said the Utah House should allow formal impeachment proceedings to begin.
The poll, which has a 3.4 percent margin of error, also showed bipartisan support for such a move, with 88 percent of Democrats and 65 percent of Republicans favoring the GOP-dominated House taking that historic step.
Of those who want lawmakers to wait, 61.5 percent said the criminal investigations should be allowed to run their course, 5 percent responded that the accusations are not serious enough, and 4 percent did not believe the allegations are true.
Swallow said legislators should wait for the facts instead of responding to polls and allegations.
"From the beginning I have asked the U.S. attorney's office to investigate so the public can know the facts rather than the implausible and impossible allegations leveled against me by people indicted or convicted of fraud," Swallow said in a statement. "The poll is not surprising but it would be disappointing if legislators act based on baseless allegations instead of waiting for the truth."
A chief deputy in the attorney general's office who rose to the top job by snagging 65 percent of the vote in November, Swallow has seen his approval rating plunge to under 12 percent in the BYU survey.
"The favorability [rating] is what I would call a historic low," Monson said.
House Speaker Becky Lockhart said Monday that public trust has eroded so much in Swallow that if a formal impeachment investigation is not needed now, it soon will be.
"We're rapidly coming to the point, if we haven't already, where the House of Representatives needs more information, and we cannot rely on the federal investigations" or other local probes to resolve allegations quickly to restore trust, the Provo Republican said during a Trib Talk appearance. "We just don't know when they will be done."
House Republicans have blocked out three hours Wednesday to discuss the impeachment process and their options for dealing with the allegations against Swallow. Lockhart said it is likely the House GOP won't make a final decision on whether to impeach Swallow during the meeting, but she expects to have a good sense of how the 61 Republican House members view the matter.
Democrats have unanimously endorsed an investigation by the House of the allegations against the attorney general, either through the appointment of a special committee or, if need be, through impeachment.
"Candidly, [the poll numbers] obviously have bearing," said House Majority Leader Brad Dee, R-Ogden.
He said the results are not surprising to him, but added that it is important to remember that voters are basing their opinions almost entirely on what the press has reported. "I'm hoping our body and I'm confident our body will be cognizant enough to weigh the results of the poll based on what information was available to those who responded."
Swallow faces investigations from federal, state and county authorities into numerous allegations, including that he helped broker deals to assist a businessman suspected of cheating customers and that he promised protection to potential donors to his predecessor, Mark Shurtleff. Swallow also faces at least two ethics complaints filed with the Utah State Bar.
This past weekend, attorneys for Marc Sessions Jenson, who is behind bars for securities violations and staring at new felony charges from the attorney general's office, alleged Swallow and Shurtleff orchestrated a shakedown, extracting more than $200,000 of favors from Jenson for themselves and others, and then prosecuting him when he failed to go along with other demands.
Swallow's attorney, Rod Snow, denied his client was involved in any such scheme. Snow argued in a letter sent to legislators last week that the House doesn't have a legal basis to impeach Swallow, because the Utah Constitution only allows for impeachment in cases of "high crimes, misdemeanors or malfeasance in office."
Snow argues that Swallow has not been charged with a crime and has not done anything ethically questionable while he was in office.
Rep. Spencer Cox, R-Fairview, who is an attorney, said Snow's reading is "blatantly wrong." The term in the Constitution comes from English law and refers to when the public trust has been lost.
"There's an old axiom in the law that if the facts are on your side, you argue the facts; if the law is on your side, you argue the law; and if neither the facts nor the law is on your side, you just argue like hell," he said. "There was a little bit of that going on."
Lockhart agreed with Cox's interpretation.
"This isn't necessarily an issue of criminality. This is an issue of public trust. We're rapidly, rapidly if we haven't already gotten to the point where we have to have more information" to restore that trust, so an impeachment investigation may be needed.
In the survey, 43 percent responded that Swallow should lose his job only if he had done something illegal and a third of those surveyed said the Republican attorney general has been a victim of the media. Fewer than 21 percent said he can still be an effective attorney general.
"It shows the public doesn't think he can do his job effectively," Monson said. "That, combined with a lot of people saying they want him to resign, those two combined show a lack of public trust."
Nine out of every 10 surveyed said the attorney general should not take campaign contributions from businesses that office is supposed to regulate. The Salt Lake Tribune reported this past weekend that Swallow accepted $105,000 in contributions from businesses that were under investigation or had already been cited by state or federal regulators.
A Tribune analysis also showed that 83 percent of Swallow's campaign donations of $5,000 or more excluding contributions from Shurtleff's political-action committee and Republican Party sources came from individuals or companies with ties to online marketing, multilevel marketing, telemarketing, payday loans or alarm companies all of which frequently draw the attention of consumer watchdogs.
The BYU survey found that 3.6 percent of those who were aware of the allegations against Swallow believe he has done nothing wrong. Thirty-four percent believe he acted illegally and 62.5 percent said he has acted unethically.
In January, after stories of the initial allegations broke questioning Swallow's conduct, 14 percent believed he had done nothing wrong and 17 percent said he had broken the law.