Weiler added the issue to the Master Study Resolution, a collection of items recommended by lawmakers to be addressed during the next nine months, leading up to the next session.
The U.S. attorney general is appointed by the president and seven states Alaska, Hawaii, Maine, New Hampshire, New Jersey, Tennessee and Wyoming plus the District of Columbia also have an appointed attorney general.
In five states, the attorney general is appointed by the governor, according to the National Association of Attorneys General.
In Tennessee, the position is appointed by the state's Supreme Court and in Maine, it is elected by the state Legislature.
"I think we should look at all those options, and I think we should look at leaving it the same," Weiler said.
"I think it's a worthy discussion."
Gov. Gary Herbert is staying out of the fray for now.
"I've heard pros and cons on it and think it's a healthy debate," he said, "but I have no position."
In a statement, Attorney General John Swallow resisted the concept of appointing the office.
"The attorney general is the guardian of the public interest and should be independent and provide legal advice based on the law instead of political pressure," Swallow said. "Utah is one of 43 states where the attorney general is elected by popular vote and this process ensures the attorney general is the lawyer for all Utah citizens."
Sen. Steve Urquhart, R-St. George, made a similar request to have the Legislature study the idea in 2011, concerned about the fundraising of Swallow's predecessor, Attorney General Mark Shurtleff.
At the time, Shurtleff called it a "lousy idea" and said it would gut the independence of the top law enforcement officer. The Legislature never explored the idea.
In 1995, a proposal was put to the Constitutional Revision Commission to have the attorney general appointed.
And in 2000, Republican Gov. Mike Leavitt pushed to strip Democratic Attorney General Jan Graham of the ability to file civil suits on behalf of the state. Eventually, an agreement was negotiated for the attorney general to consult with the governor on civil matters.
Swallow is the subject of an ongoing federal investigation exploring his involvement with Jeremy Johnson, the founder of I Works, an online coaching company, who along with his business partners is facing an 86-count indictment, including charges of money laundering and fraud.
Three Utah businessmen, speaking on condition of anonymity, have also told The Salt Lake Tribune that Swallow, when he was raising money for Shurtleff, suggested to them that contributions to the attorney general would mean special consideration if they were ever subject to a complaint to the office. Two of those men said they had been interviewed by the FBI.
Swallow raised about $1.3 million for his 2012 election victory.
His opponents in the Republican primary and general election spent less than $500,000 combined.
Changing the way Utah's attorney general is chosen would require a constitutional amendment, meaning it could not take effect until the 2016 election.
Weiler said he wants to ensure the Legislature is thoughtful and deliberate in studying the issue.
"You always want to be careful," he said, "if you're taking something away from the vote of the people."
Article VII, Section 1. (1) The elective constitutional officers of the Executive Department shall consist of Governor, Lieutenant Governor, State Auditor, State Treasurer, and Attorney General.
(2) Each officer shall:
(a) hold office for four years beginning on the first Monday of January next after their election;
(b) during their terms of office reside within the state; and
(c) perform such duties as are prescribed by this Constitution and as provided by statute.