Romanians and Bulgarians coming to the UK on New Year's Day will be welcomed by more than two-thirds of Britons if they integrate and work hard, a new poll suggests ahead of restrictions on them being lifted.

In spite of a surge of anti-immigrant rhetoric from leading politicians, British people are happy to accept migrants from the east of Europe who learn English, get a job, pay taxes and become part of their local community.

As many as 68% of those asked said they would be happy for migrants to come on those terms. That sentiment was particularly strong among people aged between 35 and 44, with 72% supporting their right to come to live and work in the UK.

The Ipsos Mori poll for the thinktank British Future comes in the wake of an intervention in the Observer by the president of Bulgaria, Rosen Plevneliev, who warned the British government not to abandon its traditional tolerance of immigrants in favour of isolation.

The business secretary, Vince Cable, responded to Plevneliev by accusing David Cameron and members of the Conservative party of adopting harmful "populist" immigration policies, such as a potential cap on EU migration and a proposed block on taking in migrants from countries with a GDP less than 75% of the UK's.

Yet, despite a barrage of negative publicity about the arrival of migrants from Romania and Bulgaria, the new poll finds that only one in four Britons (24%) believe that restricting the free movement of people, while staying in the EU, should be one of the government's priorities. A similar proportion (26%) said leaving the EU should be a priority if it does not change its rules on allowing people to come to the UK.

Nearly half (45%) said that enforcing the minimum wage was one of the most important ways of stopping business undercutting British workers by paying European workers less. Around one in five (22%) believed in the importance of managing the impact of immigration by, for example, giving more support to areas heavily affected.

The polling also showed that, while a significant majority did want a tightening of the welfare system (63%), just 2% of those asked believed that there was nothing migrants from Romania and Bulgaria could do to be accepted. This compares with 69% who said that learning the English language should be a priority for migrants, and 64% who said getting a job and paying taxes were among the key things to do.

Sunder Katwala, director of British Future, said: "I think the findings show that Romanian and Bulgarian migrants coming to work and play by the rules are welcomed; that coming to work, and not claiming before they've paid in, seems more important to people than rewriting the free movement rules or getting out of Europe, though both are legitimate long-term debates."

Meanwhile a YouGov poll of opinion leaders, organised by the all-party campaign group British Influence, also found that 81% did not feel that Cameron was talking enough about the benefits of EU membership ahead of a possible referendum in 2017.

Lord (Geoffrey) Howe, who served as chancellor and foreign secretary to Margaret Thatcher, said: "Sadly, by repeated concessions to the Eurosceptics, the government made its own position on Europe, and in Europe, more difficult."

The Bulgarian minister of labour, Hassan Ademov, told the Observer he believed the prime minister was being led into nationalistic rhetoric by the popularity of the UK Independence party. He said he did not believe there would be an influx to the UK, but revealed that the Bulgarian and British governments have agreed to work together to ensure that companies registered both in Bulgaria and the UK are prevented from exploiting the potential for cheap labour. They have also agreed to clamp down on any potential welfare fraud in a new "letter of intention" signed by both governments.

Ademov, who met employment minister Esther McVey this month to discuss the terms of the agreement, said the UK government's attitude to Bulgarian and Romanian citizens so far had been "categorically unacceptable".

He added: "My view is that the main explanation for what has been happening is that it is party election rhetoric and there is a race between the parties. God and the European commission between them have given Bulgaria and Romania the honour of having the transitional controls lifted just six months before the elections for the European parliament."