HOUSTON (Reuters) -Climate activists are prodding Exxon Mobil Corp’s top shareholders to vote against the reelection of two directors, a move that could boost a hedge fund’s proxy fight to seat four candidates on the oil giant’s board. Environmental group Sierra Club and other activists have bombarded BlackRock Inc and Vanguard Group with calls and emails urging them to vote against Exxon Chief Executive Darren Woods and lead independent director Kenneth Frazier at this month’s shareholder meeting. The two money management firms hold more than 13% of Exxon shares combined. Exxon is battling activist hedge fund Engine No. 1 over four seats on the 12-member board and the company’s future direction. The fund has criticized Exxon’s poor returns from heavy spending on fossil fuels and lack of a clear transition to cleaner fuels. It has support from some large pension funds in California and New York. BlackRock and Vanguard declined to comment on their stance. Both have signed the Net Zero Asset Managers  Initiative, vowing to press portfolio companies to achieve net zero emissions by 2050 or sooner. “The leadership of the boards of directors should be voted off” if their company fails to embrace the Paris Accord’s goal on limiting the rise in the global temperature, said Lisa Lindsley, director of investor engagement at Majority Action, one of the groups pressuring the Exxon shareholders. Exxon spokesman Casey Norton said the company supports the Paris Agreement on climate change, is focused on investments to lower carbon emissions in high-emitting sectors such as heavy industry and power, and is “the industry leader in carbon capture and storage.” Exxon is “developing innovative solutions needed to help achieve society’s goal of a lower-carbon future,” Norton said, adding that conversations with investors have been constructive. Exxon, the top U.S. oil firm, in recent months has expanded its board, pledged to increase spending on lower-carbon initiatives and lower the intensity of its oilfield greenhouse gas emissions. It has not set 2050 targets as European peers have done. In a Reuters interview last week, Woods, the CEO, said big shareholders have been receptive to Exxon’s climate efforts. Shareholders “are very encouraged by the fact that we’re taking a leadership role and helping to provide solutions to meet some of these broader ambitions,” Woods said. Supporters from various groups have sent at least 10,000 emails to both BlackRock and Vanguard and made hundreds of phone calls to BlackRock’s offices, asking them to vote against Woods and Frazier, according to the Australian-based Sunrise Project, part of a network of climate-focused groups that have zeroed in on large money managers. “It’s important to call on them to actually do more than just issue nice letters and press releases and actually make good on those commitments,” said Ben Cushing, who leads the Sierra Club’s financial advocacy campaign. Reporting by Jennifer Hiller; Editing by Leslie Adler and David Gregorio Our Standards: The Thomson Reuters Trust Principles.