House Speaker Becky Lockhart, R-Provo, has heard Seelig's request, but added that some Republican representatives want the committee to reflect the chamber's 61-14 GOP majority.
"There's been no decision on that," Lockhart said. "That will come later on when we meet with the leadership teams."
The entire House is scheduled to meet July 3 to formally create the special investigative committee, which could lead to impeachment proceedings against the Republican attorney general.
House Majority Whip Greg Hughes, R-Draper, researched how other states have conducted similar inquiries, looking at 33 investigative and impeachment committees in 11 states since 1953.
"They're kind of all over the place," he said. "If you're seeking to create a new investigative committee on its own, the prevailing makeup of that model is usually four-and-four and two co-chairs from each party."
Other states, which assigned their judiciary committee or rules committee with investigating alleged wrongdoing, were weighted in favor of the majority party.
House Republicans voted overwhelmingly Wednesday to form a special investigative committee to explore the various allegations of conflicts of interest and abuse of power against Swallow. The House becomes the fifth entity the federal, state and local investigations chief among them probing Swallow's conduct.
On Thursday, Gov. Gary Herbert said the House is moving in the right direction by beginning the investigation, adding that he recognizes the road could lead to Swallow's impeachment and ouster.
"This is a crescendo. It starts softer and will end up louder, using a musical term," Herbert said Thursday during his monthly KUED news conference. "This is beginning. It's not the end. But this beginning will tell us what the next step is."
Herbert said that, if necessary, he would "absolutely" call the Legislature into special session to establish the panel envisioned by House leaders and supported by Democrats. There is some question if the House has the legal authority to create the committee on its own.
The Republican governor said he would share information gathered by state investigators working with federal and local law enforcement, as long as it didn't interfere with the ongoing probes.
Swallow also has vowed to cooperate with the House committee.
The push for an investigative panel surprised University of Utah political science professor Matthew Burbank. He thought the House GOP would wait for the various investigations to wrap up.
"Taking this action is sort of a partial step," Burbank said. "It's not all the way to beginning impeachment, [where] you begin that investigation in a very formalized fashion, whereas this is sort of a compromise. It's sort of a half step, but it does something useful. It gives them a way to gather some more information."
But at the end of the investigative road, Burbank said, Republicans could find quicksand.
"It's hard to begin an investigation and not have some sense of where you think this is going to come out, particularly in this circumstance," he said. "More than likely, what you're going to be left with is you discover some things, but it's not clear if it makes you go to the next step of impeachment. That gray area is the real problem for Republicans. Then they have to make some very hard decisions."
He said giving Democrats equal membership on the committee would allow Republicans to argue the process is truly bipartisan. But it also would provide an equal voice to members of an opposing party who may see the information differently.
Lockhart is scheduled to meet with legislative attorneys Friday to begin discussing the panel's makeup and rules. She said she has not thought about whom she will appoint, except to say that she will not be a member.
During the Republicans' three-hour caucus Wednesday, John Fellows, the Legislature's general counsel, told lawmakers that impeachment would give them more protection from legal challenges than an investigative committee, because impeachment comes from constitutional, rather than statutory, authority.
But Lockhart said there are advantages to the investigative committee and suggested the focus may go beyond Swallow.
"With impeachment, you're looking at one specific individual, if you will, to determine if the evidence you come up with rises to the level of impeachment," she said. "An investigative committee allows you to look at all of the circumstances and all of the players involved and establish those facts."
Allegations also have been directed at Swallow's predecessor, Attorney General Mark Shurtleff.
Hughes said that, as the Legislature ventures into this new investigation, it is important to recognize that it is setting a precedent for potential future inquiries.
"The saying is: You make the rules before the game begins," he said. "Where we don't have that luxury, we need to take a step back and create a process where, if we didn't know what the roles would be and which party it would impact, it's a process that everyone could live with."