May 13 (Reuters) - CalPERS, the biggest public pension

plan in the United States, is considering voting against

ExxonMobil ( XOM ) CEO Darren Woods' re-election to the board

amid shareholder discontent over a lawsuit it filed against two

climate-focused investors, the Financial Times reported on

Friday.

Investors led by U.S. activist investment firm Arjuna

Capital and shareholder activist group Follow This earlier this

year asked Exxon and other oil companies to adopt tighter

climate targets and to set targets to reduce emissions produced

by users of its products.

In January, Exxon filed a complaint in a Texas court seeking

to prevent a climate proposal by activist investors from going

to a vote during the company's shareholder meeting in May.

Arjuna Capital and Follow This subsequently withdrew the

proposal but Exxon said in February it would continue to pursue

a lawsuit against the two activist investors.

California Public Employees' Retirement System's (CalPERS)

Chief Operating Investment Officer Michael Cohen told the FT the

pension fund was "deeply concerned" about the case, adding it

appeared to be an effort to silence critical shareholders.

"Exxon has gone well beyond any other company that we're

aware of in terms of suing shareholders for trying to bring

forward a proposal," he told the newspaper.

The fund holds a 0.2% equity stake in Exxon, based on recent

regulatory filings.

"We remain concerned about any action that could undermine

shareholders' basic rights. CalPERS encourages other investors

to join us in voicing their opposition to this lawsuit, which

attempts to silence investors," a CalPERS spokesperson said in

an emailed statement.

"The U.S. system for shareholder access is the best in the

world. To make sure it stays that way, the rules must be

enforced or the abuse by activists masquerading as shareholders

will continue threatening the system," Exxon said.

When asked if CalPERS is considering voting against Woods'

re-election as board chair, Cohen said "correct ... there are

conversations happening right now", according to the FT report.