HOUSTON—Exxon Mobil Corp. chairman and CEO Rex Tillerson will have a fight on his hands Wednesday to keep the two top jobs at the world’s biggest publicly traded oil company, as some members of the Rockefeller family and other shareholders push to separate the roles.

In what’s likely to be a raucous shareholder meeting in Dallas, much of the advance focus has been on a shareholder resolution to name an independent director as company chairman, essentially prohibiting Tillerson from holding that job and that of chief executive officer.

Tillerson, who received a compensation package last year valued at about $21.7 million, has served in both roles since 2006.

Descendants of John D. Rockefeller, the founder of Exxon Mobil predecessor Standard Oil Corp., and a variety of institutional investors in the U.S. and abroad have lined up behind the proposal, which garnered the support of 40 percent of shareholders at last year’s meeting.

Rockefeller family members and others have said they’re concerned Irving-based Exxon Mobil is too focused on short-term gains from soaring oil prices and should do more to invest in cleaner technology for the future. Separating the leadership roles, they argue, would better position the company for challenges to come.

“Exxon Mobil has long evinced a culture of defiance and indifference to many important concerns of long-term shareholders,” said New York City Comptroller William C. Thompson Jr., who oversees New York City retirement system assets. “The company can take the first step to reforming itself by separating the positions of CEO and chairman of the board.”

Foreign oil giants BP PLC and Royal Dutch Shell PLC have separate board chairs and CEOs.

But Exxon Mobil has steadfastly maintained the most effective leadership structure for the company is for Tillerson to serve in both roles.

“The board believes very strongly that there is NO single best organizational model that would be most effective in all circumstances,” the company said in its proxy statement filed with the Securities and Exchange Commission. “The board retains the authority to modify this structure to best address the company’s unique circumstances, and so advance the best interests of all shareholders, as and when appropriate.”

In its proxy, Exxon Mobil lists 17 shareholder proposals expected to be voted on at Wednesday’s meeting, several of which pertain to environmental issues such as greenhouse gas emissions and climate change. Some are proposals that have been introduced at past meetings and failed to gain a majority of shareholder support.

Some shareholders in favor of splitting the two top jobs acknowledge the company’s track record of generating good returns. Exxon Mobil posted the largest annual profit by a U.S. company—$40.6 billion—in 2007. Boosted by record crude prices to start 2008, it earned another $10.9 billion in the first three months of the year—the second-biggest U.S. quarterly corporate profit ever.