British citizens must be given priority for social housing, Conservatives will argue this week as they fight to win changes to a major new law. Labour’s Social Housing Bill goes before the House of Lords on Monday and Tories want housing providers to ensure UK citizens and people with a local connection to the area are at the “front of the queue”.

The Government’s own briefing on the Bill warns there is “now insufficient supply of good quality social housing to meet demand, with over 1.3 million households on the social housing waiting list and over 170,000 children in temporary accommodation”. Incoming Prime Minister Andy Burnham has promised his planned Number 10 North will oversee the “biggest council house building programme since the post-war period”.

Labour wants to reduce the number of council houses sold under the Right to Buy scheme and also to strengthen protections for victims of domestic abuse. But the Conservatives are pressing for safeguards for homes to go to British citizens, especially those who already live in an area, work there, have family connections or can prove service in the armed forces. The party will attempt to amend legislation going before peers on Monday so landlords “must be satisfied that the lead tenant is a British citizen”.

Sir James Cleverly, Shadow Secretary of State for Housing, Communities and Local Government, said: “British citizens should get priority when it comes to social housing. But Angela Rayner scrapped Conservative plans to put Brits at the front of the queue. More and more foreign nationals are taking up much-needed social housing as a result. Meanwhile Labour have refused to stop illegal immigrants getting social housing – which in turn is fuelling more cross-Channel migration.

“We have proposed changes to the law to ensure Brits come first, and both Labour and the Lib Dems have spoken against. Andy Burnham is promising more council houses, but unless he accepts our amendments, many of them will end up going to immigrants. But whatever he does, the Conservatives will keep fighting for fairness in the system.”

A Government spokesperson said: “Illegal migrants are not eligible for social housing. Nearly nine in ten social homes go to UK nationals and most councils have tough local connections rules in place so people must already be a resident in the area or have a connection to the local area, such as a social care worker or a nurse at a local hospital, in order to even apply for social housing.”

The homelessness charity Crisis hopes Mr Burnham will transform housing.

Matt Downie, chief executive of Crisis, said: “Andy Burnham has consistently and persuasively argued that good, affordable housing is the foundation stone on which people can build their lives and our country can build its prosperity, something we at Crisis strongly believe. As a government led by him takes shape, we hope to see that conviction turned into transformative action – and we stand ready to support.”

He added: “Everyone has the right to a safe, secure place to live. Government policy that centres on good, secure homes for all has the potential to reshape this country. We stand ready to help every step of the way.”