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 for laundering the fortunes of the ex-Soviet world’s new capitalist elite.

By then, Cyprus faced Western pressure to clean up its act, and its banks were using new software to look for suspicious customers. Mr. Manafort’s name popped into news reports after one of his clients, Ukrainian President Viktor Yanukovych, was toppled. That prompted an automated review of Mr. Manafort’s business at the Bank of Cyprus, the island’s largest commercial lender, according to three officials at the bank and Cyprus’s central bank.

Employees spent weeks forwarding the data over an encrypted line, they said, to the country’s anti-money-laundering unit, which, two years later, delivered them to the office of special counsel Robert Mueller in Washington.

For the past week, prosecutors have presented documentation of top-dollar purchases that they say constitute evidence of tax and bank fraud by Mr. Manafort. Vendors have recalled how Mr. Manafort would pay them using foreign bank wires.