ISTANBUL—President Recep Tayyip Erdogan gave official approval for a national vote on changes to Turkey’s constitution that would concentrate authority in his office.

His signing of a referendum bill on Friday set the stage for a public showdown between a leader who has steadily sought more power and detractors who say to give it to him would be dangerous.

The referendum will likely be held on April 16, Turkish officials said.

Mr. Erdogan’s supporters say the changes would streamline the government and better equip it to deal with security challenges and a sputtering economy. The government launched a massive crackdown following a failed coup attempt in July, while the country has seen an increase in attacks by the Islamic State terror group and by Kurdish separatists. Turkey’s economy contracted in the third quarter of last year for the first time since 2009.