WASHINGTON—From the earliest days of Donald Trump’s presidency, the White House knew that former senior adviser Rob Porter’s personal life could pose problems, according to administration officials.

Mr. Porter told White House counsel Don McGahn in early 2017 that his ex-wives were telling damaging stories about his behavior, which he said were untrue. The White House accepted the explanation and didn’t press for more details, waiting instead for the Federal Bureau of Investigation to finish a background investigation to determine...