Mugahed said the gunmen, riding in three 4 x 4 vehicles, ambushed a minibus and two cars of pilgrims heading along a desert road to the monastery of St. Samuel the Confessor in the Minya governorate, about 130 miles south of the Egyptian capital.
Witnesses saw between eight and 10 attackers, dressed in military uniforms and wearing masks.
The victims were travelling from the nearby province of Bani Suief to visit the monastery.
Egyptian Health Ministry Spokesman Khaled Mugahed told state television station al-Masriya that men, women, and children were among the dead and injured.
While no group has claimed responsibility for the attack, Islamic State (IS) militants have targeted Copts several times in recent months. In April, the Islamic State (ISIS) carried out two suicide bombings, leaving at least 43 dead.
Friday's shooting came after a historic visit to Egypt by Roman Catholic Pope Francis to show solidarity with the country's Christians.
"We unequivocally condemn the terrorist attack with multiple deaths that occurred today in Egypt, against Coptic Christians".
"The gunmen got on the bus and they shot people point-blank", Bishop Bakarios of the Coptic Orthodox Church in Minya Province told The New York Times, adding that numerous victims had been shot at close range.
The United Nations Security Council condemned the attack on Friday and reaffirmed "terrorism in all its forms and manifestations constitutes one of the most serious threats to global peace and security".
The grand imam of al-Azhar, Egypt's 1,000-year-old center of Islamic learning, said the attack was meant to destabilize the country.
In December, a bomb hit the main cathedral in Cairo, killing 25 people as part of what is being described as a new strategy by the Islamic State to target Christians.
The assailants reportedly stopped the group while they were traveling on a road in Egypt's Minya province.
They rallied behind el-Sissi, a general-turned-president, in 2013 when he ousted his Islamist predecessor Mohammed Morsi, who hails from the Muslim Brotherhood.
"It was a religious trip", Xinhua reporter quoted a security source at the scene as saying.
Aftermath of attack on buses and truck carrying Coptic Christians in Minya Province, Egypt, May 26, 2017.
In December past year, an attack on a Coptic church in Cairo killed 25 people.