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However, there are concerns that landlords could simply put up rent or add extra charges to folks who want to bring pets into properties that they own.
The charge for the law change is being led by Tory MP Andrew Rosindell, who said: "Sadly, pet owners moving into rented accommodation face the reality that their family could be torn apart because most landlords have unnecessary bans on pet ownership."
This week, the member of the House of Commons for Romford has been in discussions with landlords over the Dogs and Domestic Animals bill, explaining to them that it won't mean that tenants have an 'unconditional right' to own pets in their properties.
He says that anyone wanting to bring a pet into a property must pass a responsible ownership test, and there would be a checklist that must be fulfilled that includes stuff like vaccinations, microchipping, and being responsive to basic commands.
That's obviously more aimed at dog owners, isn't it? You'll have a hard time getting a boa constrictor, or even a cat, to sit or stay on your command.
Rosindell told the National Residential Landlords Association: ""The bill will include measures to ensure that pets are suitable for the type of accommodation,
"The no-pet clause means someone cannot have a dog over for even a short period for fear of recriminations or losing their home. Such discrimination must now end."
The proposed bill is being called Jasmine's law after a Weimaraner dog that was owned by the Adams family from Surrey, but was not allowed to live in a rented house.
There is growing cross-party support for the bill, and a second reading is scheduled for January 29.
Similar bills are enshrined in law in Belgium, France, Germany, and Switzerland.
However, landlord Fred Cowler said: "If it does become law, I'd want guarantees that damage and extra costs would and could be covered by the tenant."
Animal charities are right behind this bill, because it offers the opportunity to keep pets in families rather than people being forced to give them up because of restrictions.
Clare Kivlehan, from Dogs Trust, told The Mirror: "One of the main reasons for dogs handed to us is problems finding accommodation that will accept pets."