The family of former national security adviser Mike Flynn early Monday launched a legal-defense fund to help defray the former White House official’s costs associated with the investigation into Russian meddling during the 2016 U.S. presidential election.

The fund won’t disclose how much it has raised or the identities of its donors, according to a person close to Mr. Flynn. It will not accept donations from President Donald Trump’s campaign or the Trump Organization, or from the president himself, out of concern for “appearance...