The Second Amendment Foundation, a small but influential organization involved in scores of federal lawsuits to expand gun rights, is being investigated by the Washington state attorney general’s office, with the organization saying investigators have examined transactions involving its founder.

The nonprofit, its founder and top executive, Alan Gottlieb, and other related entities this week sued the attorney general’s office, saying the probe is politically motivated. State investigators seem to have shifting legal theories, the lawsuit claimed, among them that Mr. Gottlieb was “unjustly enriching himself.” In the lawsuit, the plaintiffs denied any wrongdoing.