Chaffetz, an up-and-coming GOP star, has been campaigning hard to lead the Oversight and Government Reform Committee, which has broad power to investigate, dig and probe every agency of the federal government, domestically and abroad.
He's not the most senior member of the panel, but is one of its most vocal and media-savvy, two qualities that could help him win the spot. He currently leads a subcommittee over the government's foreign operations and is a point-man on the Benghazi, Libya, terrorist attacks, having flown into the country to ask questions less than two months after a U.S. ambassador and three other Americans were killed.
"Chairman Chaffetz is fully qualified; his communication skills are among the best on the committee, best in the Congress, and he's shown genuine leadership," says Rep. Darrell Issa, a California Republican who heads the oversight committee now but is term-limited under GOP rules from continuing in the role come January.
Issa has been a sharp thorn in President Barack Obama's administration, holding hearings on issues the White House would rather avoid from the "Operation Fast and Furious" gun-selling sting to the IRS scandal involving scrutiny of tax-exempt political groups. While he could seek a rare waiver to continue heading the panel, it's unclear if the GOP brass would grant it.
Chaffetz has spent time campaigning for colleagues in several states, including the early presidential primary states of Iowa, New Hampshire and South Carolina, and raised funds to help his fellow Republicans as a way to score points with those who decide the chairmanships. The House GOP used to strictly abide by a seniority system to hand out the plum roles but now looks at the best fit, an opportunity Chaffetz is counting on.
"It's something I've had my sights set on since before I got here," he said in an interview. "I really do believe the Constitution works best when the separation of powers enables the legislative branch to oversee the executive branch. If you don't have the checks and the balances, our system falls apart."
Beyond the notoriety that comes with leading the oversight panel, the chairman also holds the power to single-handedly issue subpoenas to compel testimony or documents, a tool regularly used when the House and the presidency are held by opposing parties.
Chaffetz, a regular on Fox News, says he hopes his openness with the news media he freely hands out his cellphone number to reporters boosts his chances even if he's been in the House for only three terms.
"Other than the Cartoon Network, I think I've done them all," he says of television interviews. "If they had [a news segment], I'd get on with Tom and Jerry."
Standing in Chaffetz's way is Rep. John Mica, R-Fla., who previously chaired the Transportation and Infrastructure Committee and has been in office since 1993.
Issa, the current chairman, says both candidates would bring great qualities.
"Congressman Mica has been a chairman before and would be running on seniority, but the Steering Committee makes choices based on best suited for the job," Issa said, referring to the GOP group that consists of party leaders, key chairmen and regional representatives who select the top committee people.
The vote for committee chairs comes in November after the election determines who is still around for the coming session.
Tough job to land • Then-Rep. Jim Hansen, R-Utah, spent 20 years in the House before he headed the Natural Resources Committee for a session. No other Utah representatives in the last half-century have claimed full chairmanships. On the Senate side, Sen. Orrin Hatch has led the Judiciary and Health and Labor committees but such key spots are harder to secure in the House, with 435 members vying for the jobs.
Rep. Chris Stewart, R-Utah, lucked into a subcommittee chairmanship during his freshman year, and was then bumped up to a spot on the prized Appropriations Committee. Rep. Jim Matheson, a Utah Democrat who is leaving the House at the end of the year, serves on the Energy and Commerce Committee.
Unlike Chaffetz, Bishop hasn't been overtly campaigning to head the Natural Resources Committee that his predecessor once led.
"I don't want to feel like I'm the vulture circling around the roadkill there waiting to come into the landing," he said last week.
The current chairman, Rep. Doc Hastings, R-Wash., is retiring this year, opening up the spot. There are two other members ahead of Bishop in the pecking order: Reps. Louie Gohmert, of Texas, and Don Young of Alaska.
Young, a former chairman of the committee who is term-limited from seeking it again, is backing Bishop.
"He understands the West better than most people," Young said in an interview. "He's very articulate and this is the type of person who should lead that committee."
Gohmert is a friend, Young added, but Texas doesn't have the kind of public lands that Utah or Alaska or other states have, and that's crucial in taking on such an important role.
"The problem you have when you're not from a state like that, you can't really put a feeling on what should have happened, what could have happened and you're susceptible to having your ear bent on both sides of the aisle without understanding the real issues," Young said.
Bishop currently heads a subcommittee over public lands and environmental regulation but the full chairmanship would give him broad jurisdiction over the Interior Department and the Environmental Protection Agency. The Utah congressman says it's a prime position for a Western member.
"They own 70 percent of [Utah] and they control the other 30 [percent] through Fish and Wildlife and EPA, so, yeah, this becomes a significant issue for Utah and it's really one of the reasons I want to come back," to Washington, Bishop says. "I can use this, I think effectively, to help Utah out."
While not endorsing Bishop, the head of the Congressional Western Caucus, Rep. Steve Pearce, R-N.M., says the Utah Republican is an advocate for the issues that states in the region care about.
"Rob Bishop is a close friend, and as a member of the Natural Resources Committee has been good to the West in general and my home state of New Mexico," Pearce says. "He understands the challenges we face, which are often similar to those facing Utah."
Bring 'em on • Democrats have a slim chance to re-take the House this election, so Republicans are likely to hold on to the chairmanships they have now. If that's the case, Democrats are at least hoping they can forge good relations with the new committee leaders.
Rep. Elijah Cummings of Maryland, who as the ranking Democrat on the oversight committee has had a contentious relationship with Issa, doesn't want to weigh in on which Republican should head the panel, but he's hoping that person will want to reach across the aisle.
"Whoever they put forth, we'll do our best to try to work with for the American people," he said. "I hope they have the spirit of working together on the issues people want us to address."
That means, he says, going beyond the hot-button issues like Benghazi to looking at medicine shortages, mortgage-bank wrongdoing and credit-card hacking.
Should he take over, Chaffetz says one area that will receive renewed focus is public lands and energy issues, something that has crossover with the areas of responsibility of Bishop's committee. If the pair were each leading their respective panels, Utah would be a big beneficiary, Chaffetz says.
"There are literally thousands of bills that are introduced each year but only a couple hundred make it to the finish line," Chaffetz says. "Chairmen play a very influential role in creating that success and influencing other pieces of legislation. It's just the reality."