Foreign buyers of properties in the UK will have to pay a new levy, in a renewed attempt by Theresa May to tackle the housing crisis.

With concern growing among senior Tories that the party has allowed Brexit to drown out a compelling domestic agenda, plans unveiled on Saturday night will see foreign buyers pay extra stamp duty to fund a drive to tackle rough sleeping.

The announcement marks the start of the party’s conference in Birmingham, where the prime minister is desperate to avoid another row over her Brexit plans that might threaten to engulf her premiership. Ministers are also concerned that the party has been failing to respond to the radicalism of some of Labour’s economic programme, set out at its own conference in Liverpool last week.

On Sunday the prime minister will attempt to return to her vow to tackle social injustices and champion what she describes as the “British dream” – the idea that the next generation should do better than the last. Fixing the housing market is a major part of the programme.

It comes as the latest Opinium poll for the Observer suggest the Conservatives take a three-point lead into their conference. Despite being carried out during the Labour conference week, which can often provide a poll bounce, the poll puts the Tories on 39% support, with Labour on 36%. According to Opinium, Labour had entered its conference with a two-point lead.

The Tory plans unveiled on Sunday follow concerns that foreign buyers have been taking advantage of Britain’s housing market as a safe haven for their money, pushing up prices in the process. Party aides pointed to research from York University which estimated that 13% of new London homes were bought by non-residents between 2014 and 2016. A study by King’s College London estimated a one percentage point increase in the volume of homes being sold to overseas companies put up house prices by 2.1%.

The exact increase in stamp duty, which will be paid by individuals and companies not paying tax in the UK, will be hammered out after a consultation. Those buying a second home or buy-to-let property already pay an extra chunk of stamp duty on purchasing their property. The extra money raised will be used to support schemes in the government’s rough sleeping strategy.

Facebook Twitter Pinterest May has pledged to ‘end the scourge of rough sleeping for good’. Photograph: Dinendra Haria/Shutterstock

The plan has echoes of the new Labour policy, revealed in last week’s Observer, to increase council tax on second homes used as holiday homes. Labour said the funds raised would be used to tackle homelessness – and especially to help children in temporary accommodation.

May said: “At Conservative conference last year, I said I would dedicate my premiership to restoring the British dream, that life should be better for each new generation, and that means fixing our broken housing market. Britain will always be open to people who want to live, work and build a life here.

“However, it cannot be right that it is as easy for individuals who don’t live in the UK, as well as foreign-based companies, to buy homes as hardworking British residents.

“For too many people the dream of home ownership has become all too distant and the indignity of rough sleeping remains all too real. This government is committed to helping hardworking British residents get the right home for them and helping to end the scourge of rough sleeping for good.”

The latest Opinium poll suggests that Labour lost support among Remainers after its conference – the party was three points down among the group, with support falling from 53% to 50% of those who voted to stay in the EU.

Some Tories believe their party’s relatively healthy polling figures suggest the public were impressed with May’s recent Downing Street statement, in which she demanded more respect from the EU in its negotiations with Britain. However, there is little change in how voters regard May’s leadership qualities.

The poll does contain some good news for Labour. After its conference, it is regarded as a more united party. Last week 26% of the public perceived Labour to be generally united, which has risen to 31%. The proportion who perceive Labour to be divided has dropped from 57% to 49%.

Jeremy Corbyn’s approval rating has improved marginally, with a net approval – the difference between those who think he is doing a good or bad job – of minus 18 compared with minus 21 last week. May’s approval rating has also improved from minus 18 to minus 14. May’s lead over Corbyn on who would make the best prime minister has remained stable at 10 points.

In terms of May’s future as prime minister, 44% of the public think she should resign, which has dropped from 50% in December last year. Tory voters appear to be split between Boris Johnson (16%) and Jacob Rees-Mogg (17%) in terms of who they want as the next party leader. Opinium polled 2,008 adults online between 26 September and 28 September.

As well as concerns over its domestic agenda, there are also fears in the Tory party that its pro-business reputation is being put at risk by those pushing for a hard Brexit. A poll released by the People’s Vote campaign suggests that, while most business decision-makers regard the Tories as pro-business, their confidence falls when asked about the way in which the government has pursued the Brexit talks. Only 26% of those surveyed said the party is pursuing business-friendly Brexit policies, while 57% disagreed.

Business leaders from companies that do no trade with the EU would split 50-50 between staying in the EU and leaving, if they were given a public vote. However, among those from companies whose trade with the EU is more than 10% of turnover, the split is 70-30 in favour of staying in.

Melvin Benn, managing director of Festival Republic, which promotes Latitude, Reading and Leeds festivals, and the Download festival, said: “A hard or no-deal Brexit would not only create worrying commercial difficulties for my industry, but would cripple the economy and potentially strand our artists at passport control when they should be striding on to the main stage.”