Among other things, the investigations will examine possible meetings between Petro and drug smugglers.
There is no indication that the investigations were initiated by the White House, the newspaper writes.
However, the investigations are taking place at a time when President Donald Trump is keen to use legal cases in his foreign policy. When Venezuelan President Nicolás Maduro was captured, he was taken directly to New York to stand trial for alleged drug links.
A Trump-loyal prosecutor in South Florida has also ordered an investigation into Cuba's leader.
At the same time, Trump decided in December to pardon his ideological ally, the convicted, drug-addicted former Honduran president Juan Orlando Hernández.
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