Rice said Tuesday that such a defeat would be a serious setback for the leaders of Peru, Panama and Colombia and their efforts to strengthen democracy in their countries.
"It would send a signal loud and clear across the region that the United States cannot be trusted to keep its promises," she said in a speech at the headquarters of the Organization of American States in Washington.
Rice's speech was part of a concerted administration effort to jump-start its stalled trade agenda in the face of rising unhappiness as America's trade deficits have soared to record highs.
In the past two weeks, key House and Senate panels have approved a deal with Peru, putting it on track to be the first free trade deal passed by Congress since Democrats took control of the House and Senate at the beginning of the year.
The deal with Peru and a separate free trade agreement with Panama are given good chances of passage by Congress this year, but approval of an agreement with Colombia is seen as less likely because of concerns about human rights there.
The administration also has a free trade agreement pending before Congress with South Korea but that agreement is also given less of a chance of winning approval this year because of unhappiness with Korea's barriers to U.S. auto and beef shipments.
Rice said increased economic ties with the United States would help support the spreading movement toward democracy and free markets in Latin America.
"The exceptions to this rule may be noisy but they are heading in the opposite direction of the hemisphere," Rice said, not naming any particular country, although she later said that the United States was preparing for a transition to a democratic government in Cuba.