But there are signs that Gowdy, a former state and federal prosecutor who led the rancorous House probe into the 2012 Benghazi attacks, may defer those inquiries to other congressional investigations and to Special Counsel Robert Mueller.
"The last thing he'll want to do is impede any sort of investigation," Chaffetz said of Gowdy. "But we also have duties and obligations in the House. I trust that he'll find the proper balance to that, and it's a tricky one. It's not easy."
Gowdy's office declined requests for an interview last week, citing an ongoing review of the committee's staff and agenda. In an interview with the Wall Street Journal published Saturday, Gowdy said he saw his tenure as a "rare opportunity to depoliticize oversight" and said he had confidence in Mueller to lead the criminal probe into the Trump orbit. He said he would prefer the committee focus on issues such as the federal workforce and the coming 2020 census and drafting reform legislation.
A GOP aide acknowledged last week that Gowdy had conversations with other committee chairmen in recent weeks about their potentially overlapping jurisdiction.
"Rep. Gowdy respects the jurisdiction of each committee and has had similar conversations with all committee chairs," the aide said. "House rules clearly lay out the jurisdiction of each committee."
Any decision to bow out of probing Trump could spark a partisan battle on the Oversight committee, angering Democrats who watched Chaffetz and Gowdy vigorously pursue politically damaging probes into former Democratic presidential nominee Hillary Clinton ahead of last year's election.
Rep. Gerald E. Connolly, D-Va., the vice ranking Democrat on Oversight, said any move to have the panel step aside from Trump probes would be a "recipe for very serious friction on the committee."
"We are increasingly going to demand robust oversight on what we consider to be one of the most serious threats to American democracy," he said.
Several Republican members and aides, speaking on the condition of anonymity to describe Gowdy's thinking before he publicly unveils his oversight priorities, said the 52-year-old South Carolinian is mindful of staying in his investigative lane.
Gowdy not only conducted criminal investigations before joining Congress, but he also now sits on the House Judiciary and Intelligence committees both panels with pending oversight interests in the Trump administration. Gowdy will be wary, the Republicans said, of treading onto his colleagues' turf or interfering with Mueller's probe.
Rep. K. Michael Conaway, R-Texas, who is leading the Intelligence panel's investigation into alleged Russian election interference and possible Trump links, said last week he "had some brief conversations" on the subject with Gowdy.
"I think Trey and I will work well together," he said. "Obviously, he's got two hats, and so I trust him to be able to manage that."
Democrats have little patience for the notion that the Trump probes might be left to other committees. They pointed to multiple Obama administration issues where the House Oversight panel did its own, often higher-profile investigation of matters that were also being probed by other committees of jurisdiction.
Four House committees Intelligence, Foreign Affairs, Armed Services and Judiciary probed the Benghazi attacks, for instance. The Ways and Means Committee probed reports of the Internal Revenue Service targeting conservative nonprofit groups before then-Oversight Chairman Darrell Issa, R-Calif., took the lead, and Issa's rigorous probe into the Justice Department's handing of Operation Fast and Furious impinged on the Judiciary Committee's turf.
"We had hearings on Benghazi. We had hearings on the IRS. That never stopped us before," Connolly said. "Both Darrell Issa and Jason Chaffetz were more than willing to entertain conflict with other committees in order to engage in their own oversight and make their own imprint on the topics. So why would this to be an exception to that rule?"
Gowdy has yet to meet as chairman with the top Democrat on the Oversight panel, Rep. Elijah E. Cummings, D-Md., who has been recovering from heart surgery last month.
In a statement, Cummings said he hoped to talk to Gowdy about continuing bipartisan oversight initiatives on prescription drug prices and other issues and also about exercising the panel's "unique jurisdiction over White House officials, which includes the vetting system, the security clearance process, and the compliance with ethics rules."
Chaffetz himself emphasized the breadth of the Oversight panel's portfolio and said Gowdy will have considerably leeway to pick his targets.
"The beauty of the Oversight Committee is you have far-and-wide reach of jurisdiction," he said. "There's nothing that really holds you back. He's very collegial and will want to work closely with the other committees, but he can still do it if he wants to."
Treading on others' turf can be treacherous in terms of the internal politics of the House, however, and Gowdy is widely seen among his GOP colleagues as a team player who is unlikely to rock boats for his own aggrandizement.
Chaffetz was not always seen in that light, and his quick move earlier this year to seek Comey's memos raised some hackles, according to several members and aides.
"That's water under the bridge," Conaway said, acknowledging the tensions. "I have great confidence in Trey."
Gowdy is a conservative Republican, but he also has few personal or political ties to Trump. He endorsed Sen. Marco Rubio of Florida during the presidential race, and took heat from the Trump for the endorsement. "His hearings were a disaster," Trump said on Fox News Channel in December 2015, referring to the Benghazi probe. Earlier this year, Gowdy returned donations from a pro-Trump super PAC, and he said in the Wall Street Journal interview that he has made an assiduous attempt to keep his distance from the president.
There are pending committee inquiries into Trump matters that Gowdy now inherits. In late May, for instance, the FBI responded to Chaffetz's request for Comey's memos by citing Mueller's appointment and saying it would undertake "appropriate consultation to ensure all relevant interests implicated by your request are properly evaluated."
"How (Gowdy) deals with that, what he does, I don't yet know," Chaffetz said of the special counsel. "It's up to him."
Even if Gowdy were to give Mueller a wide berth, Democrats say he still has plenty of space to explore other Trump allegations particularly whether the president is violating his hotel lease and potentially the Constitution by continuing to own major assets while in office.
Rep. Adam B. Schiff, D-Calif., the ranking Democrat on the Intelligence Committee who also served with Gowdy on the Benghazi panel, said he hoped Gowdy would "show the same diligence about looking into this administration as they did looking into the last one" and cited the constitutional prohibition on accepting foreign gifts or "emoluments."
"There certainly appear to be violations of the emoluments clause on a pretty daily basis, and somebody needs to investigate those," Schiff said. "Those are not within the purview, for the most part, of the Intelligence Committee. They are directly under the purview of Government Reform."