"You had a one-two punch", with Mr Powell warning against policy bending to short-term political pressures and Mr Bullard saying half a percentage point would be too much, said Art Hogan, chief market strategist at National Securities in NY.
Trump on Monday tweeted that the Fed "blew it" by not cuttings rates at its meeting last week.
Kashkari said he favored a half-percentage point cut, and that he "argued we should also commit to not raising rates from the new lower level until we see core inflation sustainably reach our target".
Powell's remarks come as the central bank, which paused rate hikes this year and has suggested it could cut them in the face of concerns including the Trump administration's conflicts with trading partners, faces increasing anger from Trump.
The S&P 500 dropped 1% to 2,917, its biggest loss of the month, while the Dow Jones Industrial Average fell 179 points or 0.7%, to 26,548.
Trump has criticized the central bank for keeping credit too costly and for failing to lower interest rates last week. Some legal experts say it would be hard or impossible for Trump to remove Powell.
"We're human, we'll make mistakes - I hope not frequently - but we'll make mistakes but we won't make mistakes of integrity or character", Powell said.
Cambodia to inspect quality of under-construction buildings after fatal collapse
Rescue workers were using saws to cut steel beams and excavators to move piles of rubble of the collapsed seven-story building. He stated on Facebook that he asked Yun Min, the governor of the Preah Sihanouk province, to resign - to which he agreed.
Powell said that the Fed's independence from direct political control had served the country well and when central banks do not have that protection "you see bad things happening".
And with USA unemployment near historic lows and inflation forecast to rise only slowly, the outlook remained generally "favorable", Powell said. This will be their first face-to-face meeting after trade talks failed in May that led to increase USA tariffs on imports of Chinese goods.
Despite Tuesday's sell-off, June is shaping up to be a good month for US equities. Bloomberg reported last week that the White House had examined whether it could demote Powell, which President Donald Trump has denied.
Adding to the anxiety were Mr Trump's threats to obliterate parts of Iran if it attacked "anything American" - prompting Tehran to condemn the latest United States sanctions on Iran, and call the White House's actions "mentally retarded".
Trump's moves sent financial markets tumbling because of concerns the trade conflict could end the current 10-year economic expansion, which in July will become the longest in US history.
But risks to that outlook have grown, he added, as recent progress on resolving trade disputes had deteriorated, producing greater uncertainty. Now private economists believe from two to four rate cuts are possible this year, although some analysts think the Fed could keep policy unchanged if trade tensions are resolved without harming the economy.
Powell reiterated Tuesday that any rate changes would depend on economic data, not politics.