WASHINGTON—The Republican who leads the House committee primarily responsible for conducting oversight and investigations of the federal government said Thursday he was weighing stepping down from Congress before his term ends.

“If I do step down early, it will be months from now,” Rep. Jason Chaffetz of Utah said in an interview a day after announcing he wouldn’t run for re-election in 2018. “It’s hard for people to believe, I really do want to be with my family more. Maybe it’s more a midlife crisis more than anything else.”

Mr. Chaffetz, 50 years old, said he was exploring private-sector work and hadn’t ruled out a future run for higher office. As rumors swirled in Washington and Utah about his plans, officials in Salt Lake City began preparing for a special election to choose his successor, said Lieutenant Gov. Spencer Cox.

James Evans, chairman of the Utah Republican Party, said, “It’s my impression that Congressman Chaffetz still has a few legislative measures that he wants to shepherd through, but we are preparing in case he steps down before his term ends.”

As chairman of the Committee on Oversight and Government Reform, Mr. Chaffetz leads the House panel that investigates waste, fraud, abuse or mismanagement in the executive branch. Mr. Chaffetz promised to be an independent watchdog despite being in the same party as President Donald Trump, but many critics, including congressional Democrats, urged him to be more aggressive—particularly in investigating conflicts of interest involving the president’s businesses.