The fund will need to replace Lagarde after she was nominated last week by EU leaders as the next president of the European Central Bank, a job that will start in November.
"For Spain, and I think for all EU countries, the priority is that the director general will continue to be a European," Spanish Finance Minister Nadia Calvino said on Monday.
The issue was to be informally discussed by euro zone finance ministers at a meeting on Monday, French Finance Minister Bruno Le Maire said, urging EU states to find a common candidate as soon as possible.
No formal decision was expected as the IMF itself has not even yet opened the procedure to find a new managing director.
The head of the Washington-based IMF, whose members include most of the countries in the world, has always been a European, although in the past, large and emerging economies have challenged that practice.
The United States, despite being the IMF's largest financial backer, does not usually field a candidate because, under an informal deal with European partners, it gets the head of the World Bank - the IMF's sister organisation in the Bretton Woods system forged after World War Two.
CARNEY VS DIJSSELBLOEM?
An EU official said some EU countries wanted a candidate from an EU member state, an option that could rule out British candidates, as Britain is due to leave the EU at the end of October.
Bank of England governor Mark Carney has been mentioned in media reports as one of Europe's possible candidates.
Though born and raised in Canada, Carney, also a former governor of the Bank of Canada, holds British and Irish passports in addition to his Canadian citizenship.
Other names informally cited by officials are those of the former head of the Eurogroup, Dutchman Jeroen Dijsselbloem, and Finland's former prime minister Alexander Stubb.
When asked about Dijsselbloem, Dutch Finance Minister Wopke Hoekstra said: "The Netherlands has some excellent candidates." An official said Dijsselbloem was the only Dutch name in contention.
EU officials said the work to find a common European candidate was being led by Finland, which currently holds the EU's rotating presidency.