May 10 (Reuters) - Calpers, the biggest public pension

plan in the United States, is considering a vote against

ExxonMobil ( XOM ) chief executive Darren Woods' re-election to

the company's board in the face of 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 majors 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 that it would continue to

pursue a lawsuit against the two activist investors.

California Public Employees' Retirement System

(CalPERS)chief operating investment officer Michael Cohen told

the newspaper that 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 0.2

percent equity stake in Exxon, based on recent regulatory

filings.

Calpers did not immediately respond to Reuters request for

comment. Exxon could not be immediately reached for a comment.

When asked if Calpers is considering voting against Woods'

re-election as board chair Michael Cohen, Calpers' operations

chief said "correct", adding that "there were conversation

happening right now.