David Cameron makes his first visit to Scotland on Thursday since the independence referendum last September amid a renewed war of words between the Tories and the SNP.

As a new opinion poll suggested that the SNP could hold the balance of power at Westminster after the general election, with 55 of Scotland’s 59 seats, Downing Street confirmed that the prime minister plans to diminish the role of Scottish MPs.

On the eve of the publication of the draft clauses of the parliamentary bill that will implement the promised new powers of devolution, No 10 confirmed that the prime minister plans to block Scottish MPs from voting on some tax matters.

The dramatic change in Scottish politics after the referendum, which raises the prospects of Labour finding itself dependent on SNP votes in the event of a hung parliament, was illustrated by an Ipsos Mori/STV poll which gave the SNP a 28-point lead over Labour. STV made a projection that the SNP could take 55 out of Scotland’s 59 seats, possibly handing the nationalists the balance of power in a hung parliament. Nick Clegg formed the coalition in 2010 on the basis of 57 Liberal Democrat MPs.

The poll placed the SNP on 52% among those certain to vote, 28 points ahead of Labour on 24%. The Tories were on 12% while the Lib Dems tied with the Greens on 4%. This would give the SNP 55 seats – an increase of 49 – while Labour would see its 41 seats in Scotland shrink to four.

Following the no vote in last September’s independence referendum, the SNP has seen a surge of support under its new leader, Nicola Sturgeon. A Survation poll for the Daily Record earlier in the week suggested signs of a recovery for Scottish Labour under Jim Murphy with the gap narrowing to 20 points, but it too indicated that the SNP would be in a powerful position in Westminster after May.

The Scottish first minister Nicola Sturgeon said the SNP would seek to increase its influence at Westminster by voting on English-only matters in the House of Commons. The SNP currently refrains from voting at Westminster on education and NHS matters on the grounds that such votes do not apply to Scotland.

Sturgeon made clear that the SNP would use its projected new weight at Westminster to vote on the NHS if a future Tory government moved to privatise the health service. In an interview with the BBC’s political editor Nick Robinson, the first minister said: “If there was a vote in the House of Commons to repeal the privatisation of the health service that has been seen in England, we would vote for that because that would help to protect Scotland’s budget.”

The intervention by Sturgeon is designed to answer the central Labour warning to voters in Scotland: that a vote for the SNP could end up helping the Tories by diminishing Labour. The UK government has no say over the running of the NHS in Scotland and would therefore not be able to privatise the health service north of the border. But Sturgeon fears a stealth privatisation because the funding for the NHS in Scotland is related to funding in England.

The Tories have responded to the SNP surge by warning of the dangers of a post-election pact between Labour and the nationalists which would allow the nationalists to decide tax levels in England. The SNP have ruled out a deal with the Tories at Westminster but have said they would be open to a “confidence and supply” arrangement with Labour in which they would support the budget and the Queen’s speech in exchange for concessions. Labour is privately ruling out any deal with the SNP.

Cameron’s visit is designed to coincide with the publication of the draft clauses for the bill that will implement new devolved powers. No 10 made clear that a future Tory government would seek to block Scottish MPs from voting on some taxation matters. “The principle of English votes will apply in those circumstances,” the prime minister’s spokesman said.

George Osborne told the Treasury select committee on Tuesday that his plans are consistent with the cross-party Smith commission on further devolution which says that Scottish MPs would still vote on the UK budget. The No 10 spokesman said the plans to block Scottish MPs from voting on some taxation matters were consistent with this commitment. “There remain matters which are not devolved in which case there [are] UK-wide [votes] and there will be financial matters that will be devolved. Income tax is a very good example.”

Downing Street made clear that the prime minister would also use his visit to Scotland to say that the SNP’s fiscal policy for an independence, which assumed an oil price of around $100 a barrel, has been undermined by the recent fall in the price of oil. Asked whether the prime minister believes that the SNP’s fiscal policy has been blown apart by the fall in the price of oil: “I don’t think that is only the prime minister’s view. I think it is rather widely held by commentators and by many people up and down the country.”

Cameron will say: “Be in no doubt, whoever forms the UK government after 7 May, these new powers are guaranteed. The Scottish parliament will have more control of its tax and spending – making it one of the most powerful devolved parliaments in the world.”The all-party Smith Commission was set up in the aftermath of the referendum, when the country voted to reject independence, to fast-track a new package of powers. Cameron will add: “In September the people of Scotland came out in record numbers to decide the future of the United Kingdom. They voted clearly and decisively to keep our family of nations together. But a ‘no’ vote did not mean ‘no change’.

Insisting that the “vow”, made by all three Westminster party leaders in the final days of the referendum campaign, has been fulfilled, he will say: “The leaders of the other main political parties and I promised extensive new powers for the Scottish Parliament – a vow – with a clear process and timetable. We said a command paper would be ready by the end of October – and it was. We said we’d get cross-party agreement by St. Andrew’s Day – and we did. We said draft legislation would be published by Burns Night – and here we are, three days before the celebrations start, with those clauses before us.”

The clauses, which will be published in a Command Paper, will contain Smith Commission recommendations including giving the Scottish parliament full power to set income tax rates and bands, about £3bn of welfare powers including the housing elements of universal credit, and the devolution of fracking licensing and management of the Crown Estate in Scotland.

Westminster has already moved to allow the Scottish Parliament to extend its franchise so 16 and 17-year-olds could vote at the 2016 Holyrood elections.

Cameron will conclude: “When the people of Scotland voted to remain part of the United Kingdom, it wasn’t just about the future of devolution, it was because they valued the safety and security of being part of something bigger – our family of nations.”