LONDON—Prime Minister David Cameron and Donald Tusk, the president of the European Council, said talks on how to overhaul the U.K.’s relationship with the European Union would continue Monday after they failed to reach a comprehensive agreement on Sunday evening.

In a statement after the two men had met to thrash out a deal on Sunday, Mr. Cameron’s office said the European Commission had agreed that the U.K. qualified for a mechanism to restrict in-work benefit payments to European migrants but the two sides had not agreed on areas related to economic governance and clamping down on abuse of the free movement in the bloc.

Officials will meet early in Brussels on Monday and work through the day to resolve the outstanding issues, the statement said. Mr. Tusk had signaled that he planned to circulate a draft text with the proposals for changes to the U.K.’s relationship with the EU to member states of the bloc on Tuesday, it added.

An EU official said Mr. Tusk would assess the situation after the next day of talks and decide whether to present the proposals or not. The official declined to comment on the content of the negotiations.

Mr. Tusk, who is formally leading the negotiations for Brussels, was expected to write to EU leaders on Monday to set out his proposals for addressing the U.K. demands if the two men had reached an agreement on Sunday on the proposals to pave the way for Mr. Cameron’s planned referendum on membership of the EU.

“No deal yet. Intensive work in next 24 crucial,” Mr. Tusk said in a message on Twitter after the two men held discussions over dinner consisting of smoked salmon, followed by fillet of beef and vegetables, and finished with pear and apple crumble.

“A good meeting with @eucopresident who has agreed to another 24 hours of talks before publishing the draft UK renegotiation text,” Mr. Cameron tweeted.

Mr. Cameron has promised Britons he will negotiate a series of changes to the U.K.’s relationship with the EU before holding the referendum. No date has been set but the national plebiscite could take place as early as June if Mr. Cameron and other EU leaders agree to a deal at the next European summit in Brussels on Feb. 18.

Mr. Cameron’s most contentious demand was thought to be that EU migrants should only be eligible to receive in-work welfare payments after they have been in the country for four years. The proposal, which Mr. Cameron has argued is necessary to help stem the large influx of EU migrants coming to the U.K., has been criticized by many EU politicians for being discriminatory and undercutting the bloc’s single market for goods and services by restricting the free movement of workers.

Heading into the meeting Mr. Tusk said any agreement must be acceptable to the whole of the EU and there would be no compromise on fundamental freedoms.

Before the meeting the U.K. side had said a counterproposal for a so-called emergency brake to temporarily limit in-work welfare payments to EU migrants in Britain needed to be strengthened so it could be implemented immediately. But afterward Mr. Cameron’s office said the Commission had made clear that the U.K.’s current circumstances met the criteria for triggering the emergency brake.

“This is a significant breakthrough, meaning the prime minister can deliver on his commitment to restrict in-work benefits to EU migrants for four years,” it said. “But there are still areas where there is more to do and both agreed it was therefore worth taking the extra time to make further progress.”

On economic governance, Mr. Cameron’s office said it wanted to ensure enforcement mechanisms were watertight. The U.K. has been pushing to ensure EU countries that don’t have the euro can safeguard their rights as the single currency area integrates further.

The U.K. also wanted to see more substantive proposals on clamping down on the abuse of free movement within the bloc, including closing routes to Britain which had enabled non-EU migrants to stay in Britain in recent years, Mr. Cameron’s office said in the statement.

The debate about the U.K.’s membership of the EU is picking up steam in Britain with campaigners on both sides putting forward their case ever more forcefully as expectations grow that the referendum could take place as early as this summer. Those in favor of remaining in the EU say membership and access to Europe’s single market brings huge economic benefits and leaving would damage the U.K.’s standing in the world. Mr. Cameron has said he would campaign to stay in the EU if he can get the right deal, but if he doesn’t he has said he rules nothing out.

The euroskeptics, including many in Mr. Cameron’s Conservative Party, say the U.K. must leave the EU to regain control over a range of policy areas including migration and trade and EU legislation places an unnecessary burden on U.K. businesses. They have argued that the prime minister’s demands for a new relationship with the EU don’t go far enough.