Swallow's spokesman, Paul Murphy, said resignation is not on the table.
"The attorney general has not broken any laws and has no plans to resign," Murphy said. "We would hope these representatives would put their faith in the justice system instead of a trial by the media."
Ray's call for Swallow to step down comes amid growing concern from House members about the scandal.
House Speaker Becky Lockhart said that some lawmakers are ready to impeach the state's top cop, while others are awaiting results of the investigation.
"I'd say there are members of the House on the entire spectrum," the Provo Republican said Friday. "Some who are very frustrated and very concerned and are expressing a desire to move forward and others who desire to have a wait-and-see perspective.
"I sense there is growing frustration," she added, "as we're learning more and more information."
Against that backdrop, Lockhart emailed House members information Thursday evening laying out the process for impeachment and the options available to lawmakers should they choose to act.
It was sent in response to questions the speaker said she has been asked by House members and was intended to inform, not influence, lawmakers on the impeachment process in case the chamber exercises that option.
"It is my sincere hope that impeachment be something we never have to address," Lockhart wrote in her email. "But in the event it has to happen, we will certainly be prepared."
Rep. Jeremy Peterson, R-Ogden, said he wants to see the results of the probe before the House considers impeachment, but added that Swallow should step aside pending those results.
"I don't know what is motivating our attorney general," Peterson said. "It's frustrating for me to watch this, because it's like watching a train wreck in slow motion. You know how it's going to turn out, but we're sort of hamstrung if you want the process to do what it's supposed to do."
Peterson said he is also frustrated that the investigation, which began last year, is taking so long.
"I wish we had the facts right now. I'm tired of waiting. This is painful," he said. "Let's get to the bottom of it and either impeach or exonerate him, but let's do something, because in the meantime the attorney general's office's reputation gets torn to shreds, and it's going to take time to regain the public confidence in the office."
Impeachment, if it goes forward, would begin with articles filed in the House and, if they receive two-thirds support in that chamber, move to the Senate for a trial and potential punishment, including possible expulsion.
Rep. Brian King, D-Salt Lake City, said he believes the mood in the House on both sides of the aisle is building toward a bipartisan consensus calling for Swallow to step down.
"I don't know when that will happen, but I think that momentum is building," he said. "There are bipartisan discussions between individuals because I've been involved in them and I sense there are a number of folks with R's behind their name in the House chamber who are ready to take some action and move forward in a way that's proactive."
King said that action could range from public calls for resignation by House members, a similar statement from legislative leaders, a special session to pass a resolution calling for an investigation into whether Swallow should be impeached or the creation by leaders of a special investigative committee.
Utah Attorney General John Swallow and, increasingly, his predecessor, Mark Shurtleff, have been under scrutiny on a number of fronts:
• Swallow has been accused by indicted businessman Jeremy Johnson of helping to arrange to bribe Senate Majority Leader Harry Reid, D-Nev. Swallow says he only helped Johnson set up a lobbying deal.
• Three Utah businessmen have said Swallow, as a fundraiser for Shurtleff in 2009, suggested that a contribution to Shurtleff's campaign would get them special consideration if there were complaints about their operations to the attorney general's office.
• A complaint has been made to the Utah State Bar 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.
• 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.
• 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.
• Businessman Darl McBride provided a recording of a 2009 breakfast meeting in which Shurtleff offered him $2 million to take down a website criticizing Mark Robbins, Jenson's former business partner. Shurtleff said he could get the money from Jenson because of his plea deal. Jenson said he refused.