NICOSIA, Cyprus—The biggest bank on this island was at the tail end of a purge of its riskiest customers—from arms traders to online casinos—when its computers started combing the books to check on an American who was attracting more public attention: Paul Manafort.

It was 2015, and the future chairman of the Donald Trump presidential campaign was still a low-profile, high-earning adviser to heads of state from Eastern Europe to Central Africa. For eight years, he had banked at this small island nation, which had a big reputation...