Institutions managing about $8 trillion in assets are supporting climate-risk disclosure resolutions to be put before ExxonMobil's and Chevron's AGMs on Wednesday, May 25.
The resolutions ask both Exxon and Chevron to explain how resilient their portfolios and strategy would be if policy measures to restrict warming to 2C, as agreed in Paris in 2015, were successfully enforced.
The US Securities and Exchange Commission has ruled that Exxon must include the resolution at its AGM despite the firm arguing it already provides adequate carbon disclosures.
Carbon Tracker's James Leaton said: "Two degree scenarios need to become the new default setting for how companies report on their future business strategy - it's not clear what the oil majors are so afraid of that they resist focusing on a smaller higher margin business."
Earlier this month, 49% of shareholders backed a resolution urging Occidental Petroleum to stress test its business against the global target.
Meanwhile, Carbon Tracker says that 1000 academics from some of the world's top universities - including Oxford, Cambridge, Yale and Harvard - have publicly backed both resolutions.
A report published in March by Carbon Tracker found that while Chevron's climate disclosures generally lagged its peers it was representative of thinking right across the fossil fuel sector.
Royal Dutch Shell and French company Total, who have AGMs tomorrow, have generally been more responsive to shareholder and public pressure to take steps to align their businesses than their US counterparts, Carbon Tracker said.