Boris Johnson and former housing minister Mark Prisk are among the Tory MPs supporting a new private member’s Bill where private tenants registering for council tax would give details of the tenure and their landlord.

The Bill – whose objective is to root out bad landlords – also has the support of two Labour MPs.

The Local Government Finance (Tenure Information) Bill has been tabled by Dame Angela Watkinson.

A second private member’s Bill, the Homes (Fitness for Human Habitation) Bill tabled by Karen Buck has attracted support from MPs including – briefly – Labour housing minister Jack Dromey.

Buck, Labour MP for Westminster North, wants to amend the Landlord and Tenant Act 1985, so that residential rented accommodation must be “provided and maintained in a state of fitness for human habitation”.

Her Bill has been welcomed by Betsy Dillner, director of Generation Rent, who said: “Private renting has doubled in the past decade but legislation has not caught up with reality. Millions of people face the prospect of renting privately their whole life but they have very little protection from unscrupulous landlords and agents letting out substandard homes.

“Karen Buck is a tireless campaigner on housing and understands what needs to change for private renters. She has our full support and we hope her Bill will finally bring private renting into the 21st century.”