The likely Labour leadership contender has accused Boris Johnson of failing to take the crisis seriously, despite the announcement of £260m of pre-Christmas government cash to cut homelessness.
Sir Keir, speaking to The Independent, said the average annual increase in rough sleepers since the Conservatives came to power in 2010 was 13 per cent – taking the total from 1,768 at the start of the decade to 4,751 in 2017.
If the trend continued, he argued, the total would reach 9,914 by 2024 after five years of Mr Johnson in power, saying: “Unless the Tories take urgent action, this crisis is only going to get worse.”
Sir Keir, who is expected to launch his bid to succeed Jeremy Corbyn in the new year, called for Labour’s proposal for a £100m fund for emergency winter shelters to be adopted.
And he added: “There also needs to be a cultural shift. We have to start treating housing as a basic human right – and give councils the resources they need to provide it.”
Announcing the new funding – £134m for London boroughs, with the remainder for the rest of the country – Mr Johnson joked about sleeping out four years ago, to raise awareness.
But government figures show a 30 per cent leap in rough sleeping that year, with big rises following in 2016 (16 per cent) and the following year.
Although the official total dipped by 2 per cent last year, ministers were rapped by the UK statistics authority amid evidence that some councils had deliberately underreported numbers in their areas.
Research for the charities Homeless Link and St Mungo’s has shown that funding for homelessness services has been slashed by £1bn under the Conservatives, Labour said.
Last year, 726 people who were homeless died in England and Wales, an increase of 50 per cent in just five years.
Sir Keir, the new leadership favourite with some bookies, gave a foretaste of his likely campaign when he said: “Labour lost the election, but we didn’t lose our values.
“We must continue the moral fight against injustice, poverty and homelessness – and give people the hope that another future is possible.”
Even the official figure of 4,677 people sleeping rough last year – a snapshot of those bedded down on the street, in doorways, parks, tents and sheds on a single night – is thought to be a severe underestimation.
The homeless charity Crisis estimates that 24,000 people were rough sleeping last year and figures published by the Greater London Authority suggest 8,855 people bedding down on the capital’s streets.
Meanwhile, the number of households considered newly homeless or at risk of becoming so leapt by 11.4 per cent across England in the past year, a rise of almost 7,000 households.
The number in temporary accommodation, such as hostels and B&Bs, was already at its highest level since 2007 and rose by 1,500 to 86,130.