Violence in the city of Sohag came hours after Egyptians started voting on the new constitution, which represents a key milestone in a military-backed roadmap put in place since the removal of the Islamist Mohammed Mursi last July.
A massive security operation is underway to prevent Mr Mursi's supporters from disrupting the vote and in anticipation of attacks by Islamic militants.
The officials say police opened fire on about 300 pro-Mursi protesters after the officers came under fire from rooftops during a rally.
The new constitutional text strips out disputed Islamist language while strengthening state institutions that defied Mr Mursi: the military, the police and the judiciary
With no sign of a campaign against the constitution, the draft is expected to pass easily, backed by the many Egyptians who staged mass protests on June 30 against Mr Mursi's rule before he was deposed. The Brotherhood had called for a boycott of the vote.
Many Egyptians backing the plans cite a desire to bring stability to the country after three years of turmoil ignited by the historic uprising that felled veteran autocrat President Hosni Mubarak in 2011.
The vote that may set the stage for a presidential bid by army chief General Abdel Fattah al Sisi. His Islamist opponents say he is the mastermind of a coup that kindled the worst internal strife in Egypt's modern history and revived an oppressive police state.
But after a failed experiment with democracy, many are weary of the upheaval that has gripped this nation of 85 million and shattered its economy. They view General Sisi as someone who can stabilise the country.
Analysts say his candidacy in a presidential poll now seems a foregone conclusion as there appeared to be no alternatives. He inspected a polling station after voting began, dressed in desert coloured fatigues.