Laborde took the gold medal at last week's Pan American Judo Championships and was considered a contender for an Olympic gold in China this summer.
''We have a specific protocol for who we contact and how we go about it in these situations,'' Team USA President Jose Rodriguez, said in a Miami Herald report.
The newspaper said it was told by a source that Laborde would seek political asylum.
Rodriguez said he had contacted a government agency but did not say which one, the newspaper said.
An official with the tournament told the Herald that Laborde, 28, took her bags with her Sunday and left a note for her teammates. Contents of the note were not disclosed.
Immigration officials said they had not heard from Laborde, and the Herald said she had not been seen in the Cuban immigrant community.
The rest of the Cuban team flew home Monday.