Two trustees of the Fine Arts Museums of San Francisco, including former City Attorney Louise Renne, have quit the board in opposition to the leadership of embattled board President Diane B. Wilsey.

Renne and Dan Johnson, also an attorney, gave up their seats on the Board of Trustees, claiming they could no longer uphold their duties to protect the financial integrity of the city-owned FAMSF, which includes the de Young Museum and the Legion of Honor.

“I just felt that the governance at the museum made it virtually impossible for any board member to fulfill their fiduciary duty,” Renne said on Friday.

The resignations come in the wake of an ongoing controversy that erupted when Wilsey allegedly authorized a $450,000 disability severance payment without board approval to a former city engineer who worked at the museum. This was revealed in a whistle-blower suit by a former chief financial officer at the museum, who has since left and filed a grievance.

Wilsey has consistently claimed she did not need board approval for the payment, but the office of state Attorney General Kamala Harris launched an investigation. In April, a tentative settlement was reached in which anonymous donors would reimburse FAMSF for the $450,000 payment.

The settlement deal was the last straw for Renne and Johnson, who said he received a letter inviting him to contribute to the payoff. As a result, Johnson resigned immediately after the last board meeting, April 19, and Renne resigned shortly thereafter.

“I did not believe it was appropriate for a public entity to anonymously repay the money that was owed because it did not address the underlying problem, which was the decision to pay the money in the first place,” Johnson said on Friday.

Johnson didn’t know if the money had actually been repaid.

Board spokeswoman Amanda Duckworth would not confirm or deny that the repayment of funds had happened. “We are not going to comment on anything related to the AG (attorney general) matter,” she said by email.

The resignations of Renne and Johnson from the 43-member board follow those of Bernard Osher and Jack McDonald, who left last fall.

Also gone is Laura Hussey, the interim chief financial officer at FAMSF. Repeated attempts to reach Hussey, at both FAMSF and at her financial consulting firm, were unsuccessful.

Duckworth said that Hussey had been retained only through April 30, and had decided to stick to that date.

Hussey took over for Michelle Gutierrez, the permanent CFO who was demoted and then terminated. Gutierrez had claimed Wilsey had acted improperly in issuing a check to Bill Huggins, the retired engineer, who was also the husband of the late Therese Chen, a FAMSF registrar said to be a close confidante of Wilsey.

Wilsey never denied issuing the check, but said she had acted within FAMSF bylaws, and that she did not need board authority for the severance payment.

She also said the check was the idea of her accuser, Gutierrez, who co-signed the check. Gutierrez has retained attorney Joe Cotchett to pursue an action against FAMSF.

Repeated efforts to reach Wilsey for comment on Friday were unsuccessful.

Sam Whiting and Leah Garchik are San Francisco Chronicle staff writers. Email: swhiting@sfchronicle.com, lgarchik@sfchronicle.com