Wirth, who has spent about 35 years at Chevron, is a leading candidate to replace Watson, people familiar with the matter told the Wall Street Journal and Reuters. Sources told the Journal the decision has not been finalized and could change.
The move would put a refining specialist at the helm of the San Ramon, California-headquartered integrated oil company. Wirth holds a bachelor's degree in chemical engineering and spent roughly 10 years as executive vice president of Chevron's downstream and chemicals business — a segment responsible for refining crude oil into gasoline and operating the massive facilities that produce petrochemicals like plastics.
Refining operations have been critical to protecting the balance sheets at giant oil and gas companies at a time of persistently low oil and gas prices. Exxon Mobil, Royal Dutch Shell and Total have also elevated refining specialists to the top position, the Wall Street Journal notes.
In a statement to CNBC, Chevron said, "We cannot confirm the Journal's reporting. As a practice, we do not comment on rumor or speculation."
Wirth, 56, became vice chairman of the board of directors in February. He also serves as executive vice president for Midstream and Development, a title he has held since last year.
In his current role, Wirth is responsible for a broad portfolio that includes Chevron's business transporting fossil fuels and its unit that generates and manages power from gas-fired facilities and renewable sources. He is also in charge of Chevron's trading hubs, which buy and sell crude oil, natural gas and petroleum products.
According to Chevron's website, Wirth also plays a role in corporate strategy, business development, and policy, government and public affairs.
Wirth worked in the refining and chemicals business prior to his current role. Before that, he was president of global supply and trading from 2003 to 2006.
Having served in engineering, construction, and operations positions since joining Chevron in 1982, Wirth became president of marketing for Chevron's Asia, Middle East and Africa business in 2001. He also served on the board of directors for Caltex Australia Limited and GS Caltex Corporation, Chevron's main gasoline and diesel brands in the Asia-Pacific region.
Wirth has also served on the boards of the San Francisco 49ers Foundation and the Tiger Woods Foundation.
Correction: Michael Wirth served as executive vice president of Chevron's downstream and chemicals business. An earlier version misstated the company.