Lagarde last week announced she would step down in February, months before her term ends, triggering the race to succeed her.
EU leaders will appoint her replacement based on a recommendation by euro area finance ministers, who meet regularly in a format known as the eurogroup.
"The president of the Eurogroup, Kyriakos Pierrakakis, has called on member states to send their nominations by Friday," the spokesman said.
"The eurogroup, at its meeting on 9 November, will take stock of the nominations," he added. The eurozone finance ministers will meet in Brussels.
The frontrunners to succeed Lagarde are seen as former Dutch central bank chief Klaas Knot and Spain's Pablo Hernandez de Cos, currently head of the Bank for International Settlements.
