Kitzhaber, 67, and his live-in fiancée, Cylvia Hayes, are being investigated for possible corruption. Hayes is under scrutiny for allegedly using her relationship with Kitzhaber to land contracts for her business as a clean-energy consultant.
While senior Democrats have urged the four-term governor to step down, Kitzhaber remains steadfast.
"He was upset," Oregon Senate President Peter Courtney told reporters of a meeting with Kitzhaber on Thursday. "He was defiant. He's struggling."
"This is clearly a bizarre and unprecedented situation," Brown said in a statement.
The state attorney general's office on Thursday ordered Hayes to release emails from private accounts that discuss state business following a public records request. Hayes, through a lawyer, opposed the request. She has seven days to comply or appeal.
Questions about Hayes have swirled for months, but the pressure on Kitzhaber intensified in recent weeks after newspapers raised questions about whether Hayes reported all her income on her tax returns.
Neither the governor nor Hayes has been charged with any wrongdoing. But earlier this month, Oregon Attorney General Ellen Rosenblum said she was launching a criminal investigation.
Kitzhaber has said he and Hayes took steps to avoid conflicts of interest.
The Associated Press contributed to this report.