The European Commission will propose changing the EU rule requiring that refugees be registered in the first country where they arrive, the bloc’s top migration official said Friday.

The measure is a key element of the so-called Dublin Regulation, which the Commission has been under pressure to revise as refugees have arrived by the hundreds of thousands in Greece and Italy. Eliminating the rule, as some have advocated, would effectively shift the burden of receiving refugees from southern European countries, where most of them first arrive, to northern European ones, where most of them are heading.

The Commission is expected to officially present its proposal to reform the Dublin Regulation on March 16.

Under the current system, refugees can be sent back to the country where they were first registered, one of the reasons why many have tried not to be identified in the southern European countries. But it was already considered dead after German Chancellor Angela Merkel last August called the rule “obsolete” and suspended it for Syrians arriving in her country.

“It is our intention to change it,” EU Migration Commissioner Dimitris Avramopoulos told reporters Friday, when asked whether the rule would be done away with.

Officials said a final decision had not been made yet on whether to completely scrap the first-country-of-arrival rule, adding that it could merely be softened.

Avramopoulos’ comment came as the Greek commissioner was outlining an ambitious plan to save the free-passport Schengen area, which has come under increasing threat as several countries have reimposed border controls to halt the flow of refugees.

In its report released Friday, the Commission said the costs of losing the Schengen agreement — resulting in full reinstatement of border controls across the EU — would be between €5 billion and €18 billion per year.

The Commission has set December 2016 as its “target date” for bringing to an end internal border controls reintroduced by some member countries, including Germany and Austria.

Avramopoulos also promised progress on the EU’s plans to relocate refugees across the bloc, which have been resisted by several countries in recent months. The commissioner said that he expected “massive relocations” to begin in coming days. “Several pledges have been made in the last few days and I really count on member states to deliver,” he said.

EU countries promised last year to relocate 160,000 refugees from Greece and Italy but as of this week only 660 have been relocated, according to the Commission.

Avramopoulos and other EU officials have worked intensively in recent days to get countries in line on migration policy ahead of Monday’s summit, which will also include a meeting with Turkish Prime Minister Ahmet Davutoğlu. The commissioner said the meeting would be “decisive” for a European solution to the migration crisis.

Also ahead of Monday’s meeting, the U.N.’s refugee agency outlined its own six-point plan to help solve the refugee crisis.

The agency’s chief, Filippo Grandi, called for the full implementation of the so-called “hotspot” approach and relocation of asylum-seekers from Greece and Italy. Member countries should comply with all the EU laws on asylum, Grandi said, urging the bloc to offer those in need of protection “more safe, legal ways to travel to Europe” so that “refugees do not resort to summers and traffickers to find safety.”

Also, Greece needs more support to handle the humanitarian emergency, Grandi said, as the situation in the country is “quickly deteriorating” with some 30,000 people now stuck near the border with Macedonia.

“We are running out of time, and strong leadership and vision are urgently needed from European leaders,” Grandi said in a statement released Friday.

Barbara Surk contributed to this article.