Her words were an echo of the 2015 general election, when the Conservatives warned English voters they would be worse off under a Labour Prime Minister ‘in the pocket’ of Ms Sturgeon and Alex Salmond.
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Labour and Tory insiders believe that attack was so devastating it helped swing potentially dozens of marginal seats.
In the end the then Labour leader Ed Miliband rejected a coalition with the SNP if he became Prime Minister.
But he was criticised by his own MPs for allowing the issue to fester for weeks by failing to rule out a so-called ‘informal’ coalition deals, such as ‘confidence and supply’ in which a smaller party agrees to vote for a Budget in return for manifesto concessions.
Last night Scottish Labour sources insisted that there would be no coalition deal, informal or otherwise.
One said that there would be “no formal deal, no confidence and supply".
“We won’t concede demands from other parties. We will work with them on areas of shared concern.”
Less than an hour later, campaigning in Bolton, Mrs May said that other parties were "lining up to prop up Jeremy Corbyn.
"We’ve seen it with the Liberal Democrats, and we see it with Nicola Sturgeon’s Scottish nationalists."
She added: ""They’re very clear that they want to do everything they can to frustrate our Brexit negotiations... And it’s their tunnel vision focus on independence that actually provides uncertainty. They want to pull the strings, try to pull the strings of this election, prop up Jeremy Corbyn and provide more risk and uncertainty for the British people and that’s not in Britain’s interests."
The row erupted as MPs voted for Mrs May's snap general election in the House of Commons.
Labour backed the vote, despite polls suggesting it could lose up to 80 seats across the UK.
The SNP abstained, but two former SNP MPs Natalie McGarry and Michelle Thomson, currently sitting as independents, voted against the poll.
Both face losing their seats on June 8.
In the immediate aftermath of Conservatives shock 2015 victory Tory insiders boasted that they already planned to play the “Scotland card” at the next general election.
The campaign was notorious for its use of attack adverts, including one which depicted Alex Salmond as a pickpocket.
Ruling out a deal, Mr Corbyn rejected calls by some within his party for a "progressive alliance" with the SNP.
Speaking after a meeting of Labour's governing National Executive Committee, he said: "There will be no coalition deal with the SNP and a Labour government.
"The SNP may talk left at Westminster, but in government in Scotland it acts right. A genuinely progressive party would not refuse to introduce a 50p top rate of income tax on the richest.
"The SNP wants to break up the UK; it has no interest in making it work better. Independence would lead to turbo-charged austerity in Scotland - not progressive politics."
Calling on Scottish voters, who gave the SNP 56 out of 59 seats in 2015, to return to Labour, Mr Corbyn said: "Nicola Sturgeon is trying to convince people in Scotland that you can get rid of the Tories by voting SNP.
"She couldn't be more wrong. Only Labour or the Tories can win this election and voting Labour is the only way to remove Theresa May from office.
"If you want progressive policies that deliver social justice, fairness and equality then you should vote for the Labour Party.
"Why vote for a poor imitation, that has overseen an increase in child poverty and the biggest increase in the working poor since devolution, when you can have the real thing in the progressive Labour Party?"
It is understood the decision to rule out any coalition was agreed after Scottish Labour leader Kezia Dugdale made a ‘passionate’ case at the NEC.
Speaking earlier in Westminster, Ms Sturgeon had held the door open to a "progressive alliance" to keep Mrs May out of Number 10 if the "parliamentary arithmetic" allowed.
Asked if she would form a coalition with Labour, she said: “If the parliamentary arithmetic lent itself to the SNP being part of a progressive alliance to keep the Tories out of government, then the SNP would seek to be part of that as we said in 2015. “
But she added: "Obviously, now, and this is one of the things that is different about 2015, I'm not sure there are many people who think Labour are going to be in a position on their own or with anybody else to form a government."
In 2015, in a bid to draw a line under Tory attacks, Mr Miliband declared that he would rather not be in power than do a deal with the SNP.
Mr Miliband said he was not going to "sacrifice the future of our country" to become a Labour prime minister.
But Ms Sturgeon warned him in return that Scots would never forgive him if the Conservatives returned to Downing Street as a result of his refusal to form any kind of formal alliance.
Just days before the vote, the Labour leader set out a plan for a potential Labour minority government at Westminster in the event of a hung parliament.
Under that proposal he planned to pressure on the SNP to back a Labour government or risk a backlash if they refused.
It follows weeks of a sustained campaign by the Conservatives suggesting that English taxpayers would be worse off if the SNP propped up a coalition Labour government.