FBI agents searched the office, home and hotel room of President Donald Trump’s longtime lawyer, Michael Cohen, on Monday, seizing records including those related to a payment to a former adult-film actress, a person familiar with the matter said.

The searches were executed by the Federal Bureau of Investigation as part of a probe by the U.S. attorney’s office in Manhattan, which has opened an investigation that is being coordinated with the office of special counsel Robert Mueller, this person said.

Mr....