European Union officials said they would probe Facebook Inc.’s handling of user data on Monday, after the social network said it is investigating whether a firm linked to the 2016 Trump campaign improperly kept user’s personal data for years despite saying it had destroyed those records.

“Allegations of misuse of Facebook user data is an unacceptable violation of our citizens’ privacy rights. The European Parliament will investigate fully, calling digital platforms to account,” the parliament’s president, Antonio Tajani, said...