At least 5,889 care homes had reported a suspected outbreak of symptomatic or confirmed coronavirus, as of May 17, Number 10 confirmed.
The figure amounts to 38% of care homes in England, the spokesperson said.
In the last week, a further 343 care homes reported suspected cases.
Care home residents are among people described as "clinically vulnerable" to coronavirus, which means many are at risk of becoming critically ill if they catch the virus.
The figures come despite Transport Secretary Grant Shapps saying on Thursday that care homes were "specialists in infection control" and that the "vast majority" had made no reports of coronavirus.
The government has been heavily criticised over its record on care homes, with many questioning a policy early on in the crisis which saw coronavirus patients "seeded" into care homes when they were discharged from hospital without requiring a negative Covid-19 test result first.
Along with testing, there are also concerns with PPE, with Niall Dickson, chief executive of the NHS Confederation describing the "failure" in care homes as "nothing short of a national disgrace".
Following criticism at Prime Minister's Questions of the government's care home response, Boris Johnson pledged £600 million to help care homes with coronavirus "infection control".
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