The Department for Communities and Local Government, which is overseeing urgent testing of the material on social housing tower blocks and other high-rise buildings across the country, classifies a private building as one that does not include social housing owned by a local authority or housing association.
The overwhelming majority of the buildings that have failed tests so far have been council-owned or social housing tower blocks.
It is not known whether the two private sector buildings that have failed are residential, nor how many other samples have been submitted by private landlords. DCLG refused to confirm whether it had a backlog of tests to conduct.
While social housing buildings must conduct mandatory testing of the material, private sector owners of high-rise buildings will not be forced to do so, as BuzzFeed News reported last week. All testing of those materials is "voluntary" for the private sector.
Thousands of people in five London towers have been evacuated while authorities assess buildings clad in the same material that is believed to have contributed to the speed of the Grenfell Tower blaze, which killed at least 79 people.
DCLG, which is leading testing of cladding in England, wrote to private owners offering a free test service, and urging owners, landlords or managers of private residential blocks to "consider" conducting tests.
Bodies representing landlords and managing agents of private residential blocks, including the Association of Residential Managing Agents and the Landlords Association, said they had circulated the DCLG advice to members but did not know if any had sent cladding for testing. ARMA said it was in the process of contacting members to confirm how many had done so.
As well as residential tower blocks, hospitals and schools have also been urged by the government to send in samples of cladding.
Number 10 said the Department of Health and the Department for Education were now urging schools and hospitals to send in samples of cladding for testing. Neither department confirmed the outcome of any tests to BuzzFeed News.
Care homes, hospices and independent hospitals around the country have also been encouraged by the Care Quality Commission to review their fire safety checks in the wake of the tragedy.
A spokesperson for the CQC said it had made the recommendation as a "precautionary measure," and said he was not aware of any fire safety issues having been found.
A public inquiry into the Grenfell Tower fire will look at why combustible cladding was put on so many tower blocks across the country.