LONDON — Boris Johnson, the front-runner to replace Prime Minister Theresa May of Britain, was ordered on Wednesday to appear in court for a preliminary hearing over allegations that he lied to the public during the 2016 Brexit referendum.

A judge ruled that Mr. Johnson, a former foreign secretary under Mrs. May, must appear on a private summons to address three counts of misconduct in public office over several claims, including the suggestion that Britain paid 350 million pounds, or $440 million, every week to the European Union for the country’s membership.

The £350 million figure was a central theme of the pro-Brexit campaign, which promised to invest that amount into the nation’s health service should Britain vote to leave the bloc. The slogan was emblazoned on a Brexit campaign bus: “We send the EU £350m a week: let’s fund our NHS instead,” referring to the National Health Service.

But after Britons voted to leave the bloc by 52 percent to 48 percent, many supporters of Brexit abandoned the pledge, saying it had been a “mistake.” But opponents of Brexit argued that the campaign slogan had been a gross misuse of official statistics to deliberately mislead voters.