David Cameron has admitted defeat in his bid to stop migrant workers claiming British in-work benefits for four years during a visit to Warsaw, according to Polish officials.

But Mr Cameron told his Polish counterpart Beata Szydlo that, unless an alternative deal to curb migration was agreed, Britain could vote to leave the EU and end the automatic right of Poles to work in the UK.

The prime minister has made curbing migrant benefits the centrepiece of his effort to renegotiate Britain’s relationship with the EU, but eastern member states have strongly rejected any measures that may discriminate against their citizens living in the UK.

Polish officials say a “deeply worried” Mr Cameron made only a weak effort to sell his idea to restrict in-work benefits for migrants for four years to Ms Szydlo. “He made a halfhearted attempt,” said one official briefed on the talks. “But he knew [that the four-year curb] was off the table”.

Asked about the Polish comments, Downing Street said: “The proposal on the table is the four-year one. As the prime minister said in his speech, he is open to different ways of dealing with this issue but we do need to secure arrangements that deliver on our objectives.”

British officials confirmed that Mr Cameron had warned his Polish hosts of the dangers of a Brexit if he failed to get a good deal on migration.

The prime minister pointed out that unless Poland and other eastern European countries gave some ground — and if Britain voted to leave the EU — hundreds of thousands of their citizens would lose the right to work in the UK under the bloc’s freedom of movement rules.

“There are issues on which we do not see eye to eye … these issues will be further discussed to try to solve them together in consensus,” Ms Szydlo said after a series of meetings with Mr Cameron. “Free movement of people is a fundamental thing … we will be convincing our partners to keep this in place.”

Poland and other eastern states have led widespread opposition in the EU to Mr Cameron’s proposal to restrict benefits to migrants coming to the UK.

The curbs are the key aspect of his plan to reach an agreement with Brussels that he can sell to the British public at a planned referendum on membership of the bloc.

Brussels is looking at options for Britain to restrict EU immigration through a so-called emergency brake if it is deemed to have overwhelmed public services, potentially offering Mr Cameron an alternative to his demands to deny benefits to migrant workers.

“We fully accept the right of Great Britain to take sovereign decisions with regards to welfare policy,” she said. “There is a common direction and the future discussions in January will only take us closer to a solution.”

[Cameron] made a halfhearted attempt. But he knew [that the four-year curb] was off the table

Polish official

More than 850,000 Poles are living in Britain, the largest foreign population, making Poland an essential part of his negotiation. Warsaw is also keen to keep Britain inside the EU, and agrees with Mr Cameron on his other demands such as making the bloc more competitive and giving more power to national governments.

Mr Cameron, who had held similar talks in Romania to try to strike a deal on benefit reform, said he still thought a deal was possible.

“If we can get these reforms … we will be better off inside [the EU],” Mr Cameron said. “I feel that there is the goodwill to reach an agreement. I want to keep Britain in a reformed EU.