Republican presidential candidate Rick Perry has been criticised across the political divide over remarks he made about the head of the US central bank.
The Texas governor said on Monday he would view it as "treasonous" if Fed Chairman Ben Bernanke printed more cash between now and the 2012 election.
The White House and ex-Bush strategist Karl Rove denounced the remarks.
Mr Perry entered the race on Saturday for next year's Republican nomination to challenge President Barack Obama.
The 61-year-old, who succeeded former President George W Bush as Texas governor in 2000, has been tipped as a strong contender to be the party's nominee.
In his first three days as a presidential contender, Mr Perry has already antagonised Democrats by implying that the US military does not respect Mr Obama as commander-in-chief.
On a campaign stop in the US state of Iowa on Monday, Mr Perry said of Federal Reserve chief Mr Bernanke: "If this guy prints more money between now and the election, I don't know what y'all will do to him in Iowa, but we would treat him pretty ugly down in Texas.
"Printing more money to play politics at this particular time in American history is almost treacherous, treasonous in my opinion."
White House spokesman Jay Carney - who is also in Iowa, with President Obama on a campaign-style bus tour - said: "I certainly think threatening the Fed chairman is not a good idea.
"When you are president or running for president, you have to think about your words. The Fed's independence is important."
Karl Rove, the former chief strategist for Mr Bush, said of Mr Perry's comments that it was "not a presidential statement".
"You don't accuse the chairman of the Federal Reserve of being a traitor to his country," Mr Rove told Fox News.
Pete Wehner, a former Bush speechwriter, wrote in a comment piece: "Perry should offer a substantive critique of Bernanke's policies, not libel the man."
Previous examples of Mr Perry's blunt talking have included his suggestion in 2009 that Texans were so fed up with federal tax policies that they might consider secession from the United States.
Last year, the Texas governor described the catastrophic oil spill in the Gulf of Mexico as "an act of God".
Mr Bernanke was appointed in 2006 by Mr Bush, a Republican, and reappointed by Mr Obama.
Under Mr Bernanke, the Federal Reserve has embarked on two rounds of quantitative easing and there has been speculation among investors that he could announce a third.
Quantitative easing is when a central bank tries to stimulate the national economy by creating new electronic money to buy corporate and government debt in an effort to lower borrowing costs.