Lael Brainard Andrew Harrer | Bloomberg | Getty Images

Revelations that Federal Reserve Gov. Lael Brainard donated to Hillary Clinton's presidential campaign has created a tough situation for the central bank, at a time when Donald Trump has been accusing the Fed of making policy based on political bias.

A day after Fed Chair Janet Yellen found herself in the hot seat on Capitol Hill over the Brainard donation, Kansas City Fed President Esther George danced around the issue on Thursday in a CNBC interview. "The law allows for Fed officials to make those kind of donations. There's nothing illegal about that," George told "Squawk Box." In general, George said: "We're always sensitive to what appearances are, and that's built into our code of conduct and other things."

George acknowledged the Fed has been under increased scrutiny during the 2016 presidential campaign, as Trump repeatedly accused Yellen of keeping interest rates low to keep the economy stable to burnish President Barack Obama's legacy and help Clinton slide into the White House. "We know in today's environment there are a lot of attention on the Fed and a lot of questions about how it operates. I, for myself, try in many matters to think about these issues of appearances," George said. Federal Election Commission records show that Brainard has donated $2,700 to the Clinton campaign, the maximum allowed for personal contributions. "It is up to the individual's discretion," George said. "Lael Brainard and other Fed officials make those judgments for themselves."

During the question and answer period of Wednesday's House Financial Services Committee hearing on banking regulations, New Jersey Republican Rep. Scott Garrett pressed Yellen on the Brainard donation. Garrett also asked Yellen about what he said were media reports that Brainard was "angling for a top job" in a Clinton administration should the former secretary of state win the November election. Yellen said she has "absolutely no awareness" of those reports, and like George, stressed campaign donations by Fed officials are legal. "I don't think there's a conflict of interest there." "What's important to me is whether or not [in] decision-making, or collective decision-making, I see politics being brought to bear. … I have never seen that on the part of any of my colleagues," Yellen said. In light of the donations, Yellen said she neither asked for, nor did Brainard offer, to recuse herself from voting on monetary policy. The Fed's next meeting is scheduled for November. But it's the post-election December gathering that investors are primarily focused on as a more likely time for the second interest rate hike in the past decade.