The attack, which came on the eve of the Muslim holy month Ramadan, happened while the bus was travelling to Saint Samuel the Confessor Monastery in Maghagha, about 140 miles south of Cairo.
Witnesses said they 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.
A health ministry spokesman said the number of dead could rise further, with 25 people wounded.
The Copts United news site said three children survived the attack.
Although no one immediately said they were behind the shooting, militants linked to Islamic State are suspected.
The Egyptian affiliate of the group is waging an insurgency mainly targeted at the volatile north of the Sinai Peninsula, although there have also been attacks on the mainland.
Egypt’s Coptic Christians have become one of their main targets and the latest attack is the third against Christians in Egypt in six months.
Twin suicide bombings struck two churches on Palm Sunday, and in December, a suicide bombing targeted a Cairo church.
IS said it carried out those attacks, which left more than 75 dead and scores wounded.
Late last month, Pope Francis visited Egyptandexpressed his support for Christians of the Muslim-majority Arab nation.
After the trip, IS vowed to escalate attacks against Christians and warned Muslims to stay away from Christian gatherings and Western embassies.