BRUSSELS—A European Union proposal to resettle tens of thousands of refugees from Syria and Eritrea across Europe met with strong resistance from some governments, raising doubt about its prospects.

At the same time, another EU plan for dealing with its refugee crisis—a naval operation in the Mediterranean Sea to destroy the vessels that smuggling gangs use to transport migrants—came in for criticism from United Nations Secretary-General Ban Ki-moon.

During a visit Wednesday to the headquarters of the European Commission, the bloc’s executive, Mr. Ban said he was skeptical about any military response to Europe’s refugee problem.

But he did call the commission’s plans to resettle 40,000 Syrians and Eritreans arriving in Italy and Greece in other EU countries, and to take in a further 20,000 refugees, a “step in the right direction.”

Mr. Ban said that “I encourage EU member states to show compassion as they consider this important proposal to share their resettlement responsibilities. This can enable the European Union to address the dramatically increasing flows of people while setting an example for other regions of the world facing similar challenges.”

The plans are subject to approval by national capitals, however, many of which face growing anti-immigrant sentiment among voters and see the proposals as encroaching on their sovereignty.

Countries would absorb a certain number of refugees, calculated based on factors such as population and gross domestic product.

According to three European diplomats, France—whose support will be crucial for the proposal to pass—opposes the plans, mainly due to a domestic backlash against migration stoked by the far-right National Front.

In a statement, French Interior Minister Bernard Cazeneuve said the proposal should better take into account France’s efforts in resettling refugees so far, as well as make sure that applicants who are denied asylum are being sent back to their home countries.

Czech Prime Minister Bohuslav Sobotka said that his country “rejects the setting of mandatory quotas,” and noted that it has already accepted many refugees from Ukraine, where the separatist conflict has displaced some 1.5 million people in the east of the country.

Several other governments, including Spain, Poland, the Baltic states and Hungary, have also spoken against national quotas in the past. The U.K., Ireland and Denmark have special arrangements exempting them from such policies.

The commission denied the proposed mechanism would result in quotas and stressed that the granting of asylum remains a sovereign right of each member state.

“We never used the word quota. All it is about is solidarity. It is then up to member states to decide to how many people they grant refugee status,” EU Migration Commissioner Dimitris Avramopoulos said.

To sweeten the deal, the commission proposes that countries get a lump sum of €6,000 ($6,550) per refugee from the EU budget.

Interior ministers from all EU countries will vote on the plans June 16. Unanimity isn’t required. “If France and Germany support it, it will fly,” another European diplomat said.

The plans also include guidelines to reassure countries such as Sweden and Germany, preferred destinations for asylum seekers, that Italy and Greece will register and fingerprint all migrants arriving on their shores.

Under EU law, migrants who seek asylum have to be fingerprinted and registered in the first European country of arrival. But in recent years, Italy and Greece have allowed people to move to other EU countries without being registered.

In parallel, the EU’s border agency, Frontex, said that it had expanded the zone of operations of its Mediterranean mission, Triton, to 138 miles south of Sicily. According to Mr. Avramopoulos, Triton is now covering what Italy’s search-and-rescue mission Mare Nostrum covered last year, before it was shut down due to budgetary reasons.

Human traffickers have been taking advantage of the lawlessness in Libya to launch overcrowded boats full of desperate migrants on perilous journeys across the Mediterranean.

Italy alone expects to see as many as 200,000 migrants reach its shores this year. More than 1,600 people have perished already this year, according to aid groups.

The EU is seeking U.N. Security Council and Libyan backing for its plan to search, capture and destroy vessels used by the smuggling gangs.

Speaking in Brussels alongside European Commission President Jean-Claude Juncker, Mr. Ban said any military action would have limited effectiveness and that destroying vessels could deprive migrants of one of the limited options for survival.

Mr. Ban said he has urged governments to tackle the migration challenge in a comprehensive way, including providing legal channels for migration and action to tackle the reasons people flee their countries.

“Of course I support…strengthening the military capacity in searching and rescuing the people,” he said. “Our priority should be given to lifesaving.”

He added “there may be some other ways” to tackle the smuggling gangs.

The commission is also looking at increasing the rate of return to home countries of people who have been refused asylum or other legal forms of residence.

Only 40% of the people who should be returned are currently being sent back, according to the commission. Frontex will be involved more in organizing such returns.

The commission also seeks to sign readmission agreements with sub-Saharan countries such as Mali and Senegal by the end of 2015, to increase the return rate. That could spur North African countries, notably Morocco, to follow suit.

The commission for the past 15 years has tried to sign a readmission agreement with the government there.

The commission proposed that 24,000 refugees be moved from Italy and 16,000 from Greece—the two countries where most arrive. It limited the program to people from Syria or Eritrea because people from those countries had a high rate of winning asylum across the EU last year.

Human Rights Watch researcher Judith Sunderland called the relocation proposal positive, adding that it was “disheartening” to see so many governments come out against it. But she said that the numbers being considered are far too low.

The United Nations “called on the international community to resettle 130,000 Syrians. We think the EU alone could accommodate that,” she said.