BRUSSELS — Days after a wave of deportations of migrants arriving in Europe from Turkey, the European Union’s executive arm proposed a new quota system for members accepting asylum seekers to ease the burden on the nations confronted with an overwhelming influx.

The quotas were part of several proposals introduced Wednesday by the European Commission, the bloc’s executive arm, to address the continent’s ineffective asylum system while avoiding a backlash from member states reluctant to accept a larger number of migrants.

The proposals would create a quota mechanism to deal with exceptional situations when a country is confronted with an unmanageable crisis. An alternative would allow for the establishment of a permanent system to redistribute asylum seekers.

The second option is likely to antagonize such countries as Hungary and Slovakia, which bluntly oppose any measures to force them to take in migrants and which argue that quotas encourage migration to the Continent.

The current system, known as the Dublin Regulation, requires asylum seekers to register in the first EU country in which they arrive, and those who do not register must be sent back there if they move to another nation in the bloc.

It broke down last year because Greece, the entry point for most migrants, was unable to cope on its own, and the German chancellor, Angela Merkel, allowed hundreds of thousands of people to stay in Germany even though they had failed to register elsewhere.

One of the proposals introduced Wednesday would keep the current system mostly intact but would establish a mechanism to provide emergency relocation of asylum seekers when certain countries are facing “disproportionate pressure.’’

“Let there be no doubt: Those who need protection must continue to receive it, and they should not have to put their lives in the hands of people smugglers,’’ Frans Timmermans, the first vice president of the commission, said in a statement. “But the current system is not sustainable.’’

Timmermans said at a news conference Wednesday that the idea was to select one of the proposals that could eventually win approval from EU governments and the European Parliament.

Under a deal worked out with Turkey in March that was put into practice Monday, about 200 migrants who arrived in Greece were sent back to Turkey.

Even as leaders such as Merkel continue to press for a broader, unified European response to the migration crisis, the commission has grown increasingly reluctant to support that approach since its earlier plans to spread 160,000 migrants across the bloc were largely ignored.

The attitude toward migrants in many parts of Europe has also soured further in recent months over concerns about how to integrate newcomers and the revelations that some terrorists were able to sneak into Europe with legitimate asylum seekers.

Underlining those fears, Frontex, the bloc’s border agency, said Wednesday that two of the terrorists who carried out the Paris attacks in November had used fake Syrian papers on the Greek island of Leros before traveling to other parts of Europe.

The episode highlights “the growing threat from foreign terrorist fighters,’’ and it is “a dreadful reminder that border management also has an important security component,’’ Fabrice Leggeri, executive director of Frontex, wrote in an annual report.

In a section of the report on Syria, Frontex wrote that a “staggering number’’ of citizens of EU countries had joined the conflict there as jihadists. “Islamist extremists will exploit irregular migration flows whenever such movements fit their plans,’’ it said.