"There's a lack of accommodation [for refugees] in Greece, especially on the islands," Leggeri told reporters at a press conference.
Greece has asked Brussels to support it with more cash to pay for sea rescue, medical care and registration of refugees streaming towards the country from crisis zones in the Middle East.
Frontex confirmed that the number of refugees travelling to Greece via Turkey has increased in recent months.
Many of the new arrivals are fleeing the civil war in Syria.
But Leggeri ruled out earlier on Wednesday taking any part in military action against people smugglers planned by European leaders, including Chancellor Angela Merkel.
"It's not possible. That's not our job," he told broadcaster ZDF's morning news show.
"I have met the future commander of this mission and told him that we can co-ordinate," Leggeri continued.
"But Frontex may not and cannot take part in a military deployment" as a civilian agency, he said.
As far as Leggeri is concerned, "border control means searching for and rescuing people" - as in the recent extension to the EU's "Triton" operation, which now covers a larger area with more ships and aircraft.
European nations are currently contemplating seeking a mandate from the United Nations Security Council to attack smugglers' boats on the Libyan coast.
SEE ALSO: UN comes to Berlin for Libya peace talks