Care home leaders have accused the government of vastly underestimating the deaths of elderly people from coronavirus, as they warned the disease may be circulating in more than 50% of nursing homes and mortality is significantly higher than official figures.

Operators of several large care providers accused the government of not paying enough attention to the tragedy unfolding in residential setting across England, as figures from three of the largest chains show 620 deaths from Covid-19 in recent weeks.

Official figures released on Tuesday showed just 237 people died from coronavirus in care homes in two weeks, suggesting government figures are failing to keep up with rising numbers of deaths outside hospitals.

Four Seasons Health Care, MHA and HC-One operate almost 700 homes between them, but still account for less than 5% of facilities in England, suggesting the overall death toll in care is likely to be well in excess of 1,000. Numerous other homes are reporting deaths in double digits as concerns grow about shortages of personal protective equipment (PPE), with care home operators pleading for more government bulk buying.

There is also concern about the impact of Covid-19 patients being discharged by hospitals into care homes filled with the frail and elderly in order to free up beds. Questions are being asked about why this is happening, when the Nightingale hospital in Docklands has received only a small number of patients – reported to be just 19 at the weekend – when it has capacity for up to 2,900 intensive care beds.

Labour’s shadow health secretary, John Ashworth, joined calls on Tuesday for the government to start publishing daily death tolls from care homes, as it does from NHS hospitals.

Care leaders said up to date figures that reflect the reality on ground were increasingly essential amid signs that deaths in care are starting to accelerate as the virus spreads rapidly in homes under lockdown. Under pressure to explain why care home deaths were not being given the prominence of those in NHS hospitals, Rishi Sunak, the chancellor, told a Downing Street briefing the government was now “working with the ONS to speed up the publication of that data”.

“There is no desire not to respect what is happening in care homes,” Sunak said. But no timetable has been given for the data to be provided.

In the face of mounting criticism of their response to the spread of Covid-19 in care homes, the government has unveiled a significant expansion of testing in the sector. The health secretary, Matt Hancock, announced overnight that everyone in a care setting who was symptomatic would get a test in future and patients discharged from hospital to residential care homes would get a routine test.

Currently only the first five symptomatic patients in a care home are tested.

Hancock said all care providers would be contacted in the coming days to offer tests to their staff with symptoms.

He said: “I am deeply conscious that people in residential care are among the most vulnerable to coronavirus. We are doing everything we can to keep workers, residents and their families safe, and I am determined to ensure that everyone who needs a coronavirus test should be able to have access to one.”

One of the largest charitable providers of care in England, MHA said it suffered 89 fatalities in the last six days, bringing to 210 its total toll.

“Government rhetoric around the care sector must urgently begin to reflect the reality on the ground,” said chief executive, Sam Monaghan. “This must include accurate and timely figures about the number of people who are sadly dying in care homes. Each and every death is a tragedy.”

Care bosses said penetration of coronavirus in their homes was running at a rate of between half and two thirds, which is many times more than the latest estimate from Chris Whitty, the UK government’s chief medical adviser. He said on Monday that there were outbreaks in just one in seven care homes in England.

Jeremy Richardson, chief executive of Four Seasons Home Care which operates 191 care homes said official figures “materially understated” the crisis. He said his network had suffered at least 100 deaths from suspected or confirmed Covid-19 with around two thirds of homes reporting cases.

“From colleagues in the sector and in Four Seasons’ experience it is closer to 60% [that have Covid-19],” Richardson said. “The problem is there has been no testing for care homes for staff or residents and that’s only beginning to happen now for care home team members. The government has only been reporting deaths in hospitals so the numbers being reported now are not accurate, at best there is a significant time lag. The number of people passing away from confirmed or suspected Covid-19 is substantially under-reported in care homes.”

The order of St John Care Trust, which operates 84 care schemes in Wiltshire, Oxfordshire, Lincolnshire and Gloucestershire said on Tuesday it is anticipating “a significant step up in deaths”. Dan Hayes, chief executive, said there are now two or more cases of Covid-19 in 30% of its facilities and that deaths in the week ending 6 April were almost double the normal rate.

“It is clear [ministers] have underestimated what the impact on care homes would be,” he said. “They haven’t been paying attention to it. They are probably waiting for a degree of accuracy [about who died from Covid] they are never going to get.”

Downing Street defended its figures on care homes saying it was being “completely transparent” and that there was a time lag in collating figures from death certificates to establish causes of deaths.

“We publish daily figures on hospitals to give up to date figures on a consistent basis,” said a spokesman for Boris Johnson. “The ONS publishes separately deaths registered in a given week including deaths outside hospital. That information obviously takes a longer time to collate. But we are absolutely clear about what is and what is not included.”

Care homes also voiced growing concern that Covid-19 patients being discharged to them from hospitals could further increase deaths. The Order of St John Care Trust has written to ministers calling for the practice to be used only as a last resort unless it can be guaranteed that individuals are no longer infectious. It said current directives “will be knowingly placing those most vulnerable to dying from this virus, directly in its path”.

The family of a 70-year-old resident of a care home in Poole told the Guardian that after the NHS asked the home to take a Covid-19 patient, the patient died while two other residents were now displaying symptoms .

“Poor people like my brother are sitting ducks in relation to this,” a relative said, requesting anonymity.

Meanwhile care homes are calling for urgent action to speed the supply of protective equipment to care homes amid reports from workers that in some cases they are being denied access to kit and that managers are having to drive hundreds of miles to buy supplies.

Richardson called for the government to seize control of the supply chain for PPE by buying in bulk and distributing it through trusted providers to avoid care homes having to deal with profiteering start-up suppliers some of whom were charging 200% of normal rates.

Geoff Crocker, chairman of Bristol Care Homes, said the government and council had together supplied 300 masks, but it needed over 500 per day across the group.

“I had to drive to Birmingham last week to obtain 1,200,” he said. “We managed to buy 2,000 in the UK. We have placed orders overseas directly for a pipeline of 3,000 and for a further 1,000 from a UK manufacturer.”

The Downing Street spokesman said the government was “working round the clock” to help the elderly, including prioritising tests for care home workers and delivering 7.8m pieces of PPE to 28,000 care settings. “We are constantly trying to get more support to the social care sector but we do acknowledge that more needs to be done,” he added.