(Reuters) - A group of Republican U.S. state treasurers will withhold votes for Vanguard fund board nominees on concerns the slate is "tone-deaf" on sustainability matters, presenting a political hurdle for the top mutual fund manager as it looks to build turnout for a rare fund trustee election on Wednesday.

The officials oversee just a small fraction of Vanguard's total assets of $10.1 trillion and aren't likely to change the final outcome of the vote to elect 13 fund trustees including Vanguard CEO Salim Ramji.

However, their objections against several prominent directors show the hard political calculus facing asset managers under attack from Republican politicians even as these firms drop priorities once seen as mainstream.

Vanguard, for instance, has rolled back boardroom diversity guidance, promised a more hands-off role with banks, and left a net-zero effort.

Vanguard representatives did not respond to questions.

In a Feb 24 letter to the Pennsylvania firm, seen by Reuters, eight state finance officials led by Missouri's Vivek Malek called Vanguard's slate "tone-deaf to recent legal, legislative, and regulatory pushback" against the consideration of environmental, social and governance (ESG) factors.

In an interview, Malek cited the example of Sarah Bloom Raskin, a director since 2018 and a one-time Fed official whose climate risk concerns doomed her nomination for a top Federal Reserve post.

Malek also cited David Thomas, a director since 2021 and the president of Morehouse College, a historically Black college in Georgia, who Malek described as an outspoken advocate of diversity, equity and inclusion (DEI) initiatives. Vanguard's proxy states Thomas's "research and published works on managing diversity in the workplace are nationally renowned."

Malek said the examples suggest the board lacks ideological balance, despite the firm's recent policy changes. "To me it seems like Vanguard's leadership should be moving in a different direction," Malek said.

Malek oversees a $4.5 billion educational savings portfolio, 90% of which is invested with Vanguard. Other letter signers were Texas Comptroller Glenn Hegar, with $1.86 billion with Vanguard, and Pennsylvania Treasurer Stacy Garrity, with $6.2 billion.

Raskin and Thomas did not immediately comment.

Jeff DeMaso, who edits a newsletter for Vanguard investors, said the criticism shows how it is becoming harder for investment managers to even talk about ESG concerns with the public or with portfolio companies directly.

"There's an element right now of trying to figure out what the rules are," he said.

Vanguard's funds are grouped in 35 trusts, overseen by the same trustees. To achieve quorum, each trust must have a third of its net assets voted.

As of Feb 19 Vanguard was still seeking participation according to a securities filing. "Every vote matters" Vanguard said, urging clients to vote so it could avoid extra time and costs of more proxy solicitation work.

(Reporting by Ross Kerber; Editing by Lincoln Feast.)