Countries with Mediterranean coasts exposed to arrivals from Africa and the Middle East, such as Italy and Greece, are demanding other states take in more migrants.
Last year, asylum claims in the EU reached a level not seen since the 2015-2016 refugee crisis, ramping up long-standing tensions between member states.
The subject dominates a meeting of interior ministers in Brussels on Thursday as the bloc makes a fresh push this year to reform the asylum system.
Ahead of the talks, several ministers urged Italy to respect the so-called Dublin Regulation under which irregular migrants should be registered in the EU country where they first enter.
Italy, under its new far-right premier Giorgia Meloni, temporarily suspended the rules in December, pointing to "purely technical reasons" linked to over-worked reception centres.
Mediterranean countries like Italy argue that the rules are an excessive burden since people arriving often want to move on and live in other EU member states such as France or Germany.
"We can see the Dublin Regulation has become complex, it almost no longer works with certain countries, notably Italy," French Interior Minister Gerald Darmanin said.
A row erupted between France and Italy in November after Rome refused to allow a rescue ship called Ocean Viking, operated by a French NGO, to dock.
- Bloc strains -
Tensions have eased somewhat, but Rome is still resisting the return of migrants that landed on its soil but continued their journey northwards.
"Even before the Ocean Viking crisis that we had with Italy, Italy was taking back one in 10 people," Darmanin said. "So we obviously need to improve on this matter."
Swiss Justice Minister Elisabeth Baume-Schneider urged Italy to respect the rules.
"I am not the only one to say that we need to maintain dialogue with Italy and ask Italy to honor the Dublin pact," she insisted.
Switzerland is not an EU member but is a part of the Dublin rules.
Germany's interior minister Nancy Faeser blamed "those states that do not want to readmit" asylum seekers for the dispute.
"I will try to make sure that states are aware of their responsibility. It is the law, they are obliged to readmit. I will remind them very strongly today," she said.
France, Germany and Switzerland signed a joint statement on Wednesday with Austria, Belgium, Denmark and the Netherlands expressing their concern over asylum seekers coming to their countries from the EU countries in which they first arrived.
They called for the existing rules to be respected and "reaffirmed their commitment to structurally reform" the Dublin Regulation.
The ministers in January wanted to focus on how to speed up the process of returning undocumented migrants to their country of origin in cases where their asylum bid fails.
The Belgian government on Thursday announced a bill to reform asylum procedures that is mainly aimed at speeding up the turnover in existing reception centres and freeing up places taken up by rejected applicants.