The right to remove passports was introduced as part of a raft of new counter-terrorism laws in November aimed at curbing the number of French citizens leaving to join jihadist groups in the Middle East.
The source said the six French were "imminently" about to travel to Syria.
Their passports and identity cards have been confiscated for six months, after which the order can be renewed.
"There will be more," said Prime Minister Manuel Valls when asked about the case on Monday.
"Around 40 cases are currently under investigation" in addition to the six, the source said.