BRUSSELS—British Prime Minister David Cameron, engaged in a high-stakes attempt to change Britain’s relationship with the European Union, is at loggerheads over one of his main demands: curbing welfare benefits for other EU citizens working in the U.K. for four years.

EU leaders have voiced strong opposition to his plan, saying it would breach the fundamental principle of free movement of people within the bloc. Mr. Cameron argues the benefits freeze would help reduce migration to the U.K.

The differences will be laid bare at an EU summit in Brussels starting Thursday, where Mr. Cameron will try to convince his counterparts of the need for change. It will be the first time his proposals will be debated by European leaders as a group, marking the start of a critical couple of months of negotiations.

The British prime minister has staked his political reputation on a gamble: promising to hold a referendum on EU membership based on concessions he has yet to secure. A vote could be held as early as June, though Mr. Cameron has said only that he would call the referendum by the end of 2017.

Ahead of the summit, European Council President Donald Tusk said parts of Mr. Cameron’s proposals seem “unacceptable,” though he said he believes the leaders could strike a deal by February if Mr. Cameron makes a strong case. European Commission President Jean-Claude Juncker said the EU’s executive is ready to look for alternative ways for Mr. Cameron to achieve what he wants on migrant benefits and that he is confident the leaders will find a solution.

The British government says it is on track to get the changes it wants, including on migration, to back its case for staying in the EU. In Parliament on Wednesday, the prime minister said “we will get a good deal for the British people.”

Other EU leaders, including German Chancellor Angela Merkel, have said they would like the U.K. to remain in the EU, but not at the cost of breaching the bloc’s fundamental principles.

Mr. Cameron has publicly acknowledged the challenge he faces, particularly on the migration issue. In Brussels, the prime minister plans to tell other leaders he is open to alternatives on the four-year benefits curb, but also will make it clear it is a “crucial” issue for him, a senior EU diplomat said. Given the complexity of the issues involved, “we aren’t on the verge of a solution,” the diplomat said.

Adding to the prime minister’s challenges, efforts to recast Britain’s relationship with Brussels come as the EU has had to focus its attention elsewhere this year, from the financial crisis in Greece to the influx of refugees from Syria and the growing terrorist threat from Islamist militants.

Mr. Cameron’s timetable for reaching an agreement with Europe is slipping. Earlier this month, the prime minister conceded he wouldn’t be able to reach an agreement by the end of this year, with February now his goal.

Europe’s current problems could fuel concerns among the British public that a borderless EU enables terrorists to move freely between countries, something that Mr. Cameron is now seeking to turn to his advantage. Over recent weeks, he has begun linking arguments for remaining in the EU to national security, saying a unified Europe is more secure and effective in facing threats such as terrorism.

At home, Mr. Cameron is under pressure from some in his center-right Conservative Party and parts of the British public to reduce immigration to the U.K., including from the EU, because they say the current high levels place a strain on public services.

Recent polls show public opinion in the U.K. about EU membership remains divided, with immigration uppermost in voters’ minds. Two polls published this week indicate more people would vote to remain in the EU if the prime minister obtained the measures he wants on migration.

“Migration is a big issue and without something on that, there is a real risk people could vote to leave,” said Stephen Booth, co-director at Open Europe, an EU policy think tank.

Among potential alternatives, according to Mr. Booth, could be an “emergency brake” in which temporary restrictions on migrant benefits could be imposed if public services in the U.K. are overwhelmed by an influx of migrants. Some European leaders have previously voiced opposition to that.

Another possibility could be to make changes to out-of-work benefits. “Discrimination of workers based on nationality is a red line,” one senior EU official said. But restricting benefits for people out of work “is a little less complicated, there is more flexibility in the court’s rulings, the longer you are out of work, the more restrictions can be put in place.”

Mr. Cameron could also opt to change the U.K.’s own benefit system in a way that would make it less generous, but that would also mean restricting access to benefits for British workers as well, something the prime minister would be reluctant to do.

Mr. Cameron’s spokeswoman has said the immigration proposal remains on the table for the summit. U.K. Foreign Secretary Philip Hammond this week that there hadn’t been any viable counter proposals to Mr. Cameron’s migrants benefits plan.

A senior EU diplomat said Mr. Cameron has rejected alternative proposals that the EU had put on the table on limiting migrants, including blocking in-work benefits for migrants for six months, instead of four years.