Federal prosecutors in New York asked the Trump Organization, President Donald Trump’s flagship company, for records relating to a $130,000 payment to a former adult-film actress as part of an investigation into Mr. Trump’s personal lawyer, according to a person familiar with the matter.

The request came in connection with raids by the Federal Bureau of Investigation Monday on the office, hotel room and home of Trump lawyer Michael Cohen. Federal prosecutors in Manhattan are examining possible bank fraud by Mr. Cohen, among...