(May 28, 2008)—Exxon Mobil chairman and CEO Rex Tillerson will likely have a fight on his hands at shareholders meeting in Dallas, Wednesday.

Tillerson is trying to keep both top jobs at the world's biggest publicly traded oil company. Some members of the Rockefeller family, as well as other shareholders, are pushing to divorce the roles.

Much of the advance focus has been on a resolution to name an independent director as company chairman, essentially prohibiting Rex Tillerson from holding that job and that of CEO.

Tillerson has served in both roles since 2006.

Descendants of John D. Rockefeller, the founder of Exxon Mobil predecessor Standard Oil Corporation, as well as institutional investors in the U.S. and abroad have lined up behind the proposal.

Critics say the company is too focused on short-term gains from rising oil prices and that it should invest more in cleaner technology for the future.