BRUSSELS — European Union leaders on Thursday set a six-month deadline for deciding whether to push ahead with plans for a border guard agency that could deploy to member states unable or unwilling to manage their borders as thousands of migrants continue to arrive in Europe daily.

The border and coast guard project is chiefly aimed at protecting Europe’s external borders in countries like Greece and Italy as people fleeing conflict or poverty for better lives in Europe overwhelm coast guards and reception facilities.

In a statement released during their summit in Brussels, the leaders tasked EU ministers to “rapidly examine” the scheme and to rule on its future by June 30 at the latest, side-stepping a potentially divisive debate on Thursday over the issue.

While the leaders note that “it is indispensable to regain control over the external borders” of Europe, the idea that the agency could send personnel, ships or planes to a country even if that nation opposes the deployment is reviving old fears about a loss of national sovereignty to unelected officials in Brussels.

Countries farther north in Europe like Germany and Sweden — the preferred destinations of many migrants — are particularly keen to have tighter controls along Greece’s twisting maritime border with Turkey to ease the flow of people, as is France.

“The external borders must be protected. I’m entirely in agreement with the proposal from the European Commission on a border guard,” French President Francois Hollande told reporters.

German Chancellor Angela Merkel also underlined that she “very strongly supports” the plan.

Before chairing the summit, European Council President Donald Tusk acknowledged that the border agency is a “most controversial idea” but said that passport-free travel in the 26-nation Schengen area would be compromised if Europe’s outside borders are porous.

“Europe cannot remain vulnerable when Schengen states are not able to effectively protect their borders,” Tusk said.

The new agency would have a standing reserve force of at least 1,500 border guards that could be sent to EU border crossings within three days, rather than relying on time-consuming calls to nations for volunteers during emergencies.

But for some, the plan touches at the very heart of national identity — a country’s right to decide who or what can be deployed on its territory — and Greece, Italy, Croatia and Hungary are wary of it.