"That's something John Swallow needs to determine for himself," Herbert said. "He needs to look at what's in the best interest of his family. He needs to look at what's in the best interest of the state."
Swallow issued a written statement Tuesday night.
"I have admired the governor for speaking out that due process should be followed and I am saddened he has grown impatient with the process," Swallow said.
"What happened to me could happen to anybody," Swallow's statement added, "and I still believe we should put our faith in fairness, facts and evidence rather than allegations, rumors and speculation shared through the media."
Herbert praised the Utah House for taking a cautious, methodical approach to the "unprecedented, potentially historical" decision of whether to impeach Swallow. House Republicans will meet June 19 to discuss the impeachment process and their options for dealing with the scandal.
Impeachment, if it moves forward, would have to start with a resolution filed in the GOP-dominated House.
The Republican governor again expressed his desire to see the Swallow investigations including one that involves resources from Herbert's own Department of Public Safety wrap up quickly, one way or the other.
"Particularly when we talk about the Department of Justice, they need to be methodical, they need to be thorough, they need to take as much time as they need, but not one day longer," Herbert said. "This is a cloud over Utah, and it needs to be removed. This is a distraction for people."
Brigham Young University political scientist Kelly Patterson said Herbert has clearly reached a tipping point, where he felt like he had to take a strong position.
"There's always a point in the unfolding of these kinds of narratives that the potential damage to their own careers or public opinion of state government makes it so public officials have to start taking sides," Patterson said. "I think with the series of allegations and stories, it's reached a point where opinion leaders in the state feel like they have to speak out because the public expects them to."
Swallow, a Republican who took office in January, is the subject of a federal probe, as well as a parallel investigation by the state and two county attorneys into a series of alleged misdeeds, including aiding an attempt to bribe Senate Majority Leader Harry Reid, D-Nev., offering special treatment to campaign donors and accepting improper, undisclosed gifts.
He also is the subject of a pair of complaints to the Utah State Bar.
And the lieutenant governor's office is in the process of hiring a special counsel to investigate whether Swallow violated campaign laws by failing to disclose business interests and income.
Swallow has denied any wrongdoing, and his attorney says he and his client expect the federal investigation will not result in any charges. Meanwhile, Swallow's spokesman has said his boss has no plans to resign.
Senate Democrats on Monday called for the Legislature's Government Operations Committee to put parties under oath and gather facts relating to allegations against the attorney general.
Senate President Wayne Niederhauser, R-Sandy, said he believes that, under the Utah Constitution, it is the House that should conduct any investigation, not the Government Operations Committee, which includes senators who could be required to sit as a jury to judge articles of impeachment against Swallow.
"We do want to know what the truth is, and perhaps it is time the Legislature get involved," Niederhauser said. "But that's not through a call from the Senate. That's a call from the House."
Allegations against Swallow
Utah Attorney General John Swallow has come under scrutiny on a number of fronts:
Bribery allegation • Indicted St. George businessman Jeremy Johnson has, at times, accused Swallow of helping to arrange to bribe Senate Majority Leader Harry Reid, D-Nev. Swallow says he only helped Johnson set up a lobbying deal.
Special consideration? • Three Utah businessmen have said Swallow, as a fundraiser for his predecessor, Mark Shurtleff, in 2009, suggested that a contribution to Shurtleff's campaign would win them special consideration if there were complaints about their operations to the attorney general's office.
Rules violation?• At least two complaints have been made to the Utah State Bar, one by the state's former director of consumer protection, alleging Swallow violated attorney-client rules by discussing a consumer-protection case with a potential donor and suggesting the target meet with Shurtleff.
Withholding information? • The lieutenant governor's office is in the process of hiring a special counsel to investigate a complaint that Swallow concealed business interests on his candidate financial disclosure forms, including a company central to the Johnson deal.
Posh vacations • Convicted businessman Marc Sessions Jenson said Swallow and Shurtleff took posh vacations to his Newport Beach, Calif., villa on Jenson's dime while he was free on a plea deal with the attorney general's office. During the trips, Jenson said they pressed him for fundraising help and other financial deals.
Shurtleff's new gig
Just days after parting ways with an international law firm, former Utah Attorney General Mark Shurtleff has started his own government-relations business.
Shurtleff announced on his Facebook page that he had formed The Shurtleff Group, a Washington, D.C.-based firm, effective June 1.
On his Twitter account, Shurtleff reported over the weekend that he was attending the Republican State Leadership Committee meeting in Mackinac Island, in upstate Michigan, where he was representing the Entertainment Software Association.
ESA represents the video-game industry and contributed money to Shurtleff's campaigns for attorney general.
As a three-term attorney general, Shurtleff touted the organization's self-imposed ratings and opposed efforts to penalize retailers that sell violent video games to minors. His office joined nine other states to argue against a violent video game ban when it went before the U.S. Supreme Court.
Shurtleff said last week that he had resigned his position with the law firm Troutman Sanders, saying the travel and demands of the job were taxing on his family.
Shurtleff declined to comment on his new venture.
He is part of a wide-ranging investigation by federal, state and county authorities into allegations that he and his handpicked successor, John Swallow, abused their office.