This comes amid violent clashes between protesters and security forces in the capital Kinshasa, which have left at least 19 people dead.
The protests follow Kabila’s failure to give up power when his mandate finished at midnight on Monday.
"If things don't proceed properly, towards dialogue and an end to the democratic impasse," France will ask the European Union to consider sanctions against Kabila, said government spokesman Stephane Le Foll.
He said Foreign Minister, Jean-Marc Ayrault, raised the possibility during a cabinet meeting.
Opposition leader Etienne Tshisekedi has called for "peaceful resistance" against the 45-year-old ruler, who is showing no sign of leave power anytime soon.
In office since his father was assassinated in 2001, Kabila was re-elected in 2006 and again in 2011.
Though he is in theory constitutionally barred from seeking a third term, under a controversial recent court order, he may stay on until a successor is chosen.
New elections the norm
"The end of Mr. Kabila's mandate should normally lead to new elections, without the outgoing president standing in them," said Stephane Le Foll.
The EU already imposed travel bans and asset freezes on seven top DR Congo officials on December 12, three months after a crackdown on anti-Kabila protesters left more than 50 people dead.
The United States has also sanctioned several senior Congolese figures close to the president.
The United Nations says at least 19 are dead and 45 injured after the latest wave of violence.