Longtime National Rifle Association leader Wayne LaPierre has told the group’s board he is being extorted and pressured to resign by the organization’s president, Oliver North, over allegations of financial improprieties, in a battle stirring up one of the nation’s most powerful nonprofit political groups.

In a letter sent to NRA board members late Thursday afternoon, Mr. LaPierre, the group’s CEO and executive vice president, said he refused the demand. Instead he called on board members to “see this for what it is: a threat...