Raqqa Is Being Slaughtered Silently, a group which has been reporting on the situation inside the city since IS took control, says the militants have stopped believers from fleeing.
Christians cannot leave "under any circumstances", the organisation said.
It has raised concerns jihadis could be planning to kill the remaining families, thought to be a few hundred people.
Most of the Christians still in the city were too poor to flee when the militants arrived.
Armenians have also been banned from leaving.
Islamic State has killed thousands of people as it takes over large parts of Iraq and Syria.
The group has forced Christians to flee areas it controls with an ultimatum to pay a tax, leave or face death.