Launching the Conservatives' Scottish manifesto in Edinburgh, Mrs May said she was committed to facing up to the challenge of meeting the care needs of an ageing population while ensuring fairness for younger generations.
Labour say the new winter fuel policy, replacing the "triple lock" pension scheme and means testing pensioners for care in the home, are a "triple whammy for pensioners".
Shadow Chancellor John McDonnell said the money raising plans by the Conservatives would "hit millions of those on low and middle incomes".
The party says plans to scrap the triple lock on pensions, end universal winter fuel payments and overhaul the social care system are an assault on the living standards of the over-65s.
She said proposals set out in the Tory manifesto for the pension age to reflect life expectancy mean that 34 million people will have to work longer if Mrs May stays in Downing Street.
He said: "This means 10 million pensioners waking up this morning to the fact they could lose their allowance".
Ms Long-Bailey said that in 2015, growth had been so weak the basic state pension would have been uprated by just £1.36 under the double lock formula.
Under current Conservative proposals, he said there were net tax rises of £14.4bn - the equivalent of £760 per family - on the horizon between now and 2021/2022.
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"Those people who worked hard, they did everything that was asked of them, they put their blood, sweat and tears into making Britain great".
"We also have 1.7 million pensioners in this country living in poverty, a million of them in fuel poverty...30,000 excess deaths every winter as a result of fuel poverty and basically people not being able to heat their homes".
Mr McDonnell also suggested it was "fair" that millionaires such as Rolling Stone Sir Mick Jagger were entitled to the handout.
"It is disgraceful and we are calling on the Conservative Party to withdraw it today".
The 65-year-old said he claimed his own winter fuel handout, saying: 'With regard to the winter fuel allowance, I spend it on winter fuel.
Speaking in the wake of the Conservative manifesto launch in the Labour-held seat of Halifax, Mr McDonnell brushed-off suggestions that his party could see significant losses in strongholds like Yorkshire.
It promises to means test the payment, "focusing assistance on the least well-off pensioners, who are most at risk of fuel poverty". "His numbers don't add up and the only way to pay for his ideas is massive tax hikes on working people".
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The 72-year-old attorney was a former FBI director before Comey and has a long history with investigations and public service. On Wednesday, Mr Trump was quoted in a White House statement as saying: "I look forward to this matter concluding quickly".
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