BRUSSELS—EU leaders on Thursday agreed to speed up discussions on sweeping new proposals to reinforce the bloc’s border protection and pressured Turkey to stick to its part of a migrant deal.

A proposal for EU border guard corps ran into little initial resistance as leaders steered clear of details of a plan that could allow Brussels to deploy people to secure the bloc’s external border without the approval of the national governments. Leaders set a six-month deadline to agree to the proposal’s fine print, including the issue of sovereignty that remains problematic for several countries.

“We are failing to protect our external borders. That is why leaders have decided to speed up on all these issues,” said the host of the meeting, European Council President Donald Tusk. “The leaders’ broad acceptance means that in the future Europe will not remain vulnerable because the [external] border is insufficiently protected,” he added.

Poland, Malta and Greece were the only nations to raise the sovereignty issue but said they were broadly supportive of the plan.

The new border guard proposal was put forward earlier this week, partly in reaction to Greece’s reluctance in accepting EU assistance in policing its sea border with Turkey.

Greek Prime Minister Alexis Tsipras said the border guard is a good idea but he still needed to identify its value compared with the Greek coast guard, according to two participants in the debate.

Several leaders complained about Turkey not sticking to its part of a migrant deal struck late November, diplomats said.

Earlier in the day, a group of 10 EU leaders, led by German Chancellor Angela Merkel and her Austrian counterpart, Werner Faymann, pressed Turkish Prime Minister Ahmet Davutoglu on the need for his government to stem the flow of migrants.

More than 800,000 refugees and other migrants have come to Europe via Turkey this year, making it the main gateway for asylum seekers arriving on the continent.

Since the EU deal with Turkey came into force in recent weeks, there has been only a “slight reduction” in the flow of migrants, the Luxembourg government, which holds the EU’s six-month rotating presidency, wrote in a paper ahead of the meeting. An average 4,000 people are arriving in Greek shores daily from Turkey, down from between 5,000 and 6,000 a day in November, the paper said.

“This decrease may, however, also be attributed to other factors,” such as colder weather, it added.

French President François Hollande warned Turkey that Europe would only honor its part of the deal if the Ankara government lowered the numbers of migrants arriving in Europe.

“With Turkey, we have put in place a certain number of rules. They need to be respected,” Mr. Hollande said on his way into the summit of all 28 EU leaders. “If we can’t control the external borders, we won’t be able to go any further in what we promised Turkey.”

As part of the November deal, EU leaders has offered Ankara €3 billion ($3.26 billion) in aid and promised to accelerate talks to bring Turkey into the EU in return for Turkey’s help in cracking down on illegal migration.

“So far there’s still much to do, on both sides,” said Swedish Prime Minister Stefan Löfvén. “Both Turkey and the EU need to deliver.”

Ms. Merkel has sought to persuade EU leaders to offer to take Syrian refugees directly from Turkey as a way to settle them more evenly across Europe instead of having them cross European borders illegally to get to Germany and other top migrant destinations. But the plan has so far failed to gain much traction, even among Ms. Merkel’s closer allies.

“We need to have the influx of migrants decrease before we can take people from Turkey,” Dutch Prime Minister Mark Rutte said after a smaller meeting of leaders ahead of the summit. The EU is willing to help Ankara with housing Syrians, Mr. Rutte said, in reference to the €3 billion that is intended to go to social projects targeting refugees. “But the number of people coming through Turkey needs to drastically go down.”

Ms. Merkel said that the European Commission would report back in February on how Turkey is implementing its side of the deal and that if the flow of migrants dropped sharply, several European countries may take groups of refugees directly out of Turkey on a voluntary basis.

Mr. Davutoglu said that the promise of EU money, the flurry of meetings with EU leaders and the prospect of visa-free travel for Turkish citizens have bolstered political support for him and the deal back home, according to two participants at the meeting. But he insisted that the EU money needed to start flowing and that European governments had to begin taking refugees directly from Turkey.

“Of course, we have differences in opinion, we voice them candidly to each other,” Mr. Davutoglu said after the meeting. “But these differences in opinion don’t constitute an obstacle.”

He said that Turkey would hold high-level meetings with Germany in Berlin on Jan. 22 and with Greece in Izmir, Turkey, in the first half of February as part of a push to coordinate trilateral efforts to halt irregular migration.