Richard Fisher, the president of the Dallas Federal Reserve, is most worried over the euro zone's second-largest economy, rather than its periphery.
In an interview with German newspaper Handelsblatt, Fisher said he was cautiously optimistic about the euro zone and sees an improvement in the crisis-hit region. But problems remained in France in particular, he said.
"I am concerned about France, more than any other country,"the Dallas Fed president told the newspaper. A strong France was important in the context of European history, he said.
"The risks to the development of the economy there must not be underestimated," Fisher said.
Recent indicators have shown that the economic picture for France appears to have improved slightly, although a sharp drop in industrial output numbers released on Friday showed a recovery is still far off. INSEE statistics agency said output shrank 1.4 percent in June from May. A Reuters poll had forecast an increase of 0.1 percent.
France's finance minister has said recently said the country exited recession in the second quarter of 2013 with growth of 0.2 percent. The figures will be confirmed on Wednesday. The Bank of France is forecasting growth of 0.1 percent for the year, although most economists expect the economy to shrink slightly this year.
Unemployment in the country was at a record 10.4 percent in the first quarter of 2013. A survey released earlier this week showed 84 percent of French people believe President Francois Hollande will not be able to cut the unemployment rate by the end of the year.
Giles Keating, head of research at Credit Suisse said he thought Fisher's comments were "a bit out of date".
"We all know that France has enormous problems. We also know that they have actually begun to address them," he told CNBC. "They have passed a labor market law. Some of the data look as though they are bottoming out."
France was no "paragon of labor market flexibility", he said, but argued things were better now than they were 12 months ago.