Social housing applicants must be forced to reveal their nationality to keep a closer watch on the number of council lets taken up by foreign nationals, immigration campaigners said.
Fewer than half of applications for council properties in London include the nationality of the tenant, Migration Watch UK has estimated, but this information is only disclosed voluntarily.
Migration Watch UK wants stricter controls on who receives social housing
The independent think-tank has called for the nationality question to be made compulsory to improve transparency over who is receiving social housing.
Sir Andrew Green, Migration Watch UK chairman, said: "It is deeply unsatisfactory that the information on new lets should be so chaotic given the huge importance of this issue to so many families.
"The government must make the nationality question compulsory. This could provide the basis for a renewed debate on the criteria by which social housing should be allocated."
The government has faced criticism for placing too much focus on its target to cut net migration outside the EU to below 100,000 by 2015.
Shadow home secretary Yvette Cooper recently said the government was failing to cut the most damaging forms of immigration as illegal entry is not affected by the key target.
Meanwhile, the Institute for Public Policy Research (IPPR) said the focus should shift away from migration to integration, including the impacts of migration on housing.
In 2011, there was a recorded housing stock of around 3.3 million in London of which 781,000 were social houses - about 404,000 local authority-owned properties and 376,000 housing association properties.
The waiting list for social housing in London is 10 times the number of new lets each year, Migration Watch UK said, with a fifth occupied by foreign nationals, which specifically does not include those born overseas who have become British citizens.
A third of new lets by London's local authorities are not recorded in the continuous recording of sales and letting, known as CORE data, while others submit data but without the nationality of new tenants.
Commenting on the report, MPs Frank Field and Nicholas Soames, co-chairmen of the cross-party group on balanced migration, said: "This is a huge issue for many people. The government must now launch a full inquiry into what is going on in the allocation of social housing in London."
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