Directors of American International Group Inc. are trying to determine whether to blame—and possibly replace—Chief Executive Peter Hancock following a major setback in the insurance giant’s turnaround plan, people familiar with the matter said.

Mr. Hancock’s 15 fellow directors are expected to debate at an early March board meeting various potential actions, including cutting his bonus or removing him, some people said. Some cautioned it is premature to predict any particular outcome at this point, and that the board will look broadly at management’s role.

The discussion marks a sign of lessened board support for the 58-year-old Mr. Hancock, who has run the giant insurer for more than two years. It is fallout from a $3.04 billion fourth-quarter loss, one of AIG’s worst results since the U.S. government bailed it out during the financial crisis.

The stock fell 9% on the Feb. 14 earnings announcement and additional bad news the next day: lowered targets for improving two closely watched profit measures.

Directors want to know “how did this happen, and who, if anyone, should be held accountable for this?” one person said.

In response to a request for comment from Mr. Hancock and the board’s chairman, Douglas Steenland, AIG said that “AIG’s Board of Directors and management team” are focused on their “clearly defined transformation plan for the company to deliver high quality, sustainable earnings.”

The statement added, “At this point every year, we actively review our past and future expected performance against our plan, and this year is no exception.”

Mr. Hancock faces a defined timetable for improving AIG’s profitability. Amid calls in late 2015 by billionaire investors John Paulson and Carl Icahn for a breakup of the company, he and the board laid out steps in January 2016 to bridge the company’s big gap in performance with peers. They set a deadline of December 2017. Mr. Icahn also called for Mr. Hancock’s firing.

Those pledges were key in the activists’ decision to pass on a board fight last year, and last spring AIG added Mr. Paulson and a representative of Mr. Icahn to the board.

AIG’s board is large, and views are expected to vary, the people said. Of the 16 board members, Mr. Hancock is the only member of management, and the others are all classified as independent directors.

The board in March also may discuss whether one of the directors would replace Mr. Hancock as interim CEO, one person said. Some members have insurance-industry experience.

The steep slide in AIG’s shares on Feb. 15 represented the stock’s biggest one-day percentage decline since 2011. AIG was the day’s biggest loser in the S&P 500. The stock has regained some ground, but at Friday’s close remained 4.9% below the Feb. 14 closing price.

AIG’s fourth-quarter loss was driven by a larger-than-expected $5.6 billion boost to claims reserves for its core business of selling property-casualty policies to businesses world-wide. The company said the charge involved an array of liability policies such as medical malpractice coverage. Some were issued more than a decade ago, but a full $3 billion of the boost covers policies from 2011 to 2015. Mr. Hancock, a former bank chief financial and risk officer, ran the property-casualty unit from 2011 to 2014, before his promotion to CEO.

The reserve boost followed a $3.6 billion increase for reserves in 2015’s fourth quarter, when some investors thought AIG had brought them to an appropriate, conservative level.

The huge additional reserve bolstering raises questions “that AIG simply didn’t adequately understand its book of business,” Keefe, Bruyette & Woods analyst Meyer Shields said.

AIG has said that most of the claims costs reflected in the fourth-quarter reserve charge will be covered by a reinsurance agreement that the insurer announced in January with Warren Buffett’s Berkshire Hathaway Inc.

AIG averted a possible bankruptcy brought on by other problems in 2008 thanks to its nearly $185 billion federal bailout. AIG’s current challenges with money-losing policies appear to stem at least partly from its comeback attempt after its near collapse. Then, some rivals complained that AIG was undercutting them on price as a way to keep customers from fleeing. AIG fully repaid taxpayers by late 2012, then focused on trying to close its profitability gap with rivals.

Over the past year, Mr. Hancock’s team has sold operations and assets deemed noncore, cut expenses and exited some problematic insurance lines while scaling back others, shedding billions of dollars of liability-insurance business that was unprofitable or not profitable enough. At the same time, Mr. Hancock has spent heavily to upgrade the company’s information-technology systems.

AIG also promised investors last year that it would to return $25 billion to shareholders by this year’s end, through stock-buyback programs and dividends. Between January 2016 through early this month, it had returned $14.3 billion.

Write to Joann S. Lublin at joann.lublin@wsj.com, Leslie Scism at leslie.scism@wsj.com and David Benoit at david.benoit@wsj.com