The attackers arrived in three pick-up trucks and opened fire on the vehicles carrying visitors to the Saint Samuel Monastery in the Minya province, about 220km south of the capital, Cairo, before fleeing the scene.
Egyptian security and medical officials told the Associated Press news agency that the death toll in the shooting has risen to 28.
The fatalities include many children, AP reported.
There was no immediate claim of responsibility.
Pictures of a bus at the scene aired by state TV showed its windows shot out.
"They used automatic weapons," Essam el-Bedawi, Minya governor, told state media.
Security forces launched a hunt for the attackers, setting up dozens of checkpoints and patrols on the desert road.
Following the attack, President Abdel Fattah el-Sisi called for a meeting with security officials.
Egypt's Christian minority, which makes up about 10 percent of the country's population, has repeatedly been targeted by armed groups.
In April, at least 45 people were killed and more than 100 wounded in two separate suicide bomb attacks on churches in the cities of Tanta and Alexandria during Palm Sunday ceremonies.
The attacks were claimed by the ISIS terrorist group.
Following the Palm Sunday bombings, Sisi declared a nationwide three-month state of emergency.
A bombing at Cairo's largest Coptic cathedral killed at least 25 people and wounded 49 in December 2016, including many women and children.