Seven states and the District of Columbia currently have an appointed attorney general, although Washington, D.C., is moving to an elected model. Most have the governor make the appointment, although one is appointed by the Legislature, the other by the state Supreme Court.
Weiler raised the issue amid questions swirling around Swallow and his predecessor, Mark Shurtleff, over whether their aggressive fundraising may have influenced decisions in the attorney general's office.
Swallow amassed $1.3 million for his 2012 race; his Democratic opponent spent $67,000. By comparison, the winner of the state treasurer's race raised $23,000 and the winner of the state auditor's race spent less than $83,000.
Weiler said it's hard to explain why one statewide office would be taking in so much more money.
"I can't reconcile that," Weiler said, "unless the donors are expecting, maybe, that person can offer them something in return."
Some lawmakers expressed concern about making such a change.
Sen. Margaret Dayton, R-Orem, worries about giving more power to the governor. Rep. Janice Fisher, D-West Valley City, said such clout should go to voters rather than yielding it to the Legislature or the governor.
Brian Tarbet, general counsel for the Utah attorney general's office, said states are moving away from an appointed attorney general and noted that issues of fundraising are not unique to the office. He said county attorneys and other prosecutors face similar issues.
"The attorney general enforces the Constitution and statutes and policies. That's our charge," Tarbet said. "You talk about appointing an attorney general. I'm not sure you want to point to the federal system, in light of our experience with independent prosecutors and special counsels, as if that's any panacea."
Sen. Jim Dabakis, D-Salt Lake City, who is also chairman of the Utah Democratic Party, chided lawmakers for being afraid to address the Swallow saga head-on and urged the committee to hold hearings on issues arising from the scandal, including limits on campaign cash.
But Sen. Scott Jenkins, R-Plain City, said Dabakis has the ability to propose solutions rather than complain about problems.
"Put your money where your mouth is," Jenkins said. "I'm sick and tired of hearing you bloviate on so many of these issues and do nothing, so I'm saying buck up here and see what you've got to offer."
During the past session, Democrats proposed legislation on campaign contribution caps, which Dabakis backed, but none of the bills received a hearing.
House Speaker Becky Lockhart, R-Provo, had previously said she anticipated the potential of impeaching Swallow would be discussed Wednesday, and while there was no formal meeting on the topic, representatives did weigh in.
Rep. Kraig Powell, R-Heber City, said lawmakers were feeling pressure from their constituents to do something, and he would like to know more about the mechanics of an impeachment process.
Powell, who worked for a lawmaker in Illinois who was impeached, said, "What I want to know is what is the standard in Utah, whether it's different from Illinois or other states."
Senate Majority Leader Ralph Okerlund, R-Monroe, said senators did not discuss the matter in their closed caucus. Senate President Wayne Niederhauser, R-Sandy, said members of the body should not discuss in case they have to sit as a jury in an impeachment.
"As one who would preside over that trial," Niederhauser said, "I take that very seriously."