David Cameron will make a heartfelt plea to the people of Scotland on Friday to vote for continued membership of the UK as he issues a strong warning against the dangers of complacency about the results of the referendum in seven months' time.

Amid jitters in the pro-UK camp that opinion polls may be shifting in the direction of independence before the vote on 18 September, the prime minister will visit the scene of Team GB's success at the London 2012 Olympics to declare: "We want you to stay."

Cameron's speech, described by the SNP deputy first minister Nicola Sturgeon as a "tawdry bid to use the Olympic Games as a political tool", is designed to change the tone of the Better Together pro-UK campaign which has been criticised for being relentlessly negative.

Cameron will hail the success of the Edinburgh-born Sir Chris Hoy, the six-time Olympic champion who was Team GB's flag carrier at the opening ceremony of London 2012, as an example of the vital contribution Scotland makes to the UK. "We would be deeply diminished without Scotland," the prime minister will say. "If we lost Scotland, if the UK changed, we would rip the rug from under our own reputation. The plain fact is we matter more in the world together."

Cameron, who will talk in personal terms of his family's ties to Scotland, aims to reach out to Scots by saying that people in England, Wales and Northern Ireland would be heartbroken if the UK was broken up. The prime minister's great-great-grandfather, Ewen, left Scotland in the mid nineteenth century to establish a business career in the Far East. Cameron's great-grandfather, Ewen Alan Cameron, lived at Blairmore Castle in Aberdeenshire.

Highlighting the deep connections between Scotland and the rest of the UK, the prime minister will say: "So to everyone in England, Wales and Northern Ireland – everyone, like me, who cares about the United Kingdom – I want to say this: you don't have a vote, but you do have a voice. Those voting are our friends, neighbours and family.

"You do have an influence. Let the message ring out, from Manchester to Motherwell, from Pembrokeshire to Perth, from Belfast to Bute, from us to the people of Scotland – let the message be this: We want you to stay."

The prime minister's speech is modelled on the approach adopted by United Canada which overturned a lead by the pro-independence Quebec campaign in the 1995 referendum after a strong appeal to the French-speaking province of the benefits of remaining part of Canada. The independence referendum was defeated by 50.58% to 49.42%.

He will say: "Sometimes, we can forget just how big our reputation is, that the world over the letters 'UK' stand for unique, brilliant, creative, eccentric, ingenious. We come as a brand – a powerful brand. If we lost Scotland, if the UK changed, we would rip the rug from under our own reputation. The plain fact is we matter more in the world together."

The prime minister will temper his emotional message by warning people across the UK of dangers of complacency. There was alarm in the pro-UK camp when a Scotland on Sunday/ICM poll on 26 January found a five point increase (from 32% to 37%) in support for independence since September. It also found a five point fall (from 49% to 44%) in support for the pro-UK camp.

The fear is that momentum may be building behind Alex Salmond as Scottish voters digest the white paper on independence published late last year by the Scottish government.

The prime minister will say: "There can be no complacency about the result of this referendum. The outcome is still up in the air and we have just seven months to go – seven months to do all we can to keep our United Kingdom as one, seven months to save the most extraordinary country in history. And we must do whatever it takes."

The SNP condemned the prime minister for a "shameful" attempt to politicise the Olympics. Nicola Sturgeon said: "This is a cowardly speech from a prime minister who uses the Olympic Park in London to give high-handed lectures against Scotland's independence but hasn't got the guts to come to Scotland or anywhere else to make his case in a head to head debate. David Cameron, as the Tory Prime Minister, is the very embodiment of the democratic case for a Yes vote for an independent Scotland – and he knows it. A Yes vote will put Scotland's future in Scotland's hands, and will mean we will never again have to endure Tory governments, prime ministers and policies we didn't vote for.

"...Unfortunately, with this tawdry bid to use the Olympic Games – past and present – as a political tool, David Cameron has exposed the utter hypocrisy and double standards at the heart of the No campaign."

Andrew Tyrie, the Conservative chairman of the Treasury select committee, warned that Salmond's plan to form a sterling currency union with the remainder of the UK would be rejected by Westminster. Speaking during a Commons debate on the referendum, Tyrie said that London and Edinburgh would have to conduct "intensive and intrusive oversight" of each other's budgets.

Tyrie told MPs: "A British monetary union would need something dramatically tougher than the eurozone rules – so tough that, on both sides of the border, if it was fully explained, I am confident our respective electorates would not want it.

"The Bank of England and the Treasury would have the power to direct a large part of Scottish economic and financial policy. For example, the Scots would probably be required to seek their approval before they could borrow in order to build schools and hospitals."

Downing Street stressed that the prime minister will visit Scotland though he will continue to decline to take part in a television debate with Salmond.