The government is expected to back making stealing cats and dogs a specific criminal offence in England and Northern Ireland.
A bill being debated in the Commons on Friday would create an offence of "pet abduction".
Under the legislation, anyone convicted of the offence could face a fine or a maximum of five years in prison.
Ministers are understood to be "very sympathetic" to the plan.
Currently, pets are considered in law to be property and stealing a pet is covered by the 1968 Theft Act. The law is similar in Scotland.
The government promised to make the abduction of dogs a criminal offence in 2021, but no such legislation has yet been passed.
Measures were included in the Animal Welfare (Kept Animals) Bill, which was dropped by ministers in May 2023, prompting a backlash from campaigners.
The initial pledge followed recommendations made by a pet theft taskforce established in response to concerns about dog theft during the Covid pandemic.
Then-home secretary Priti Patel said: "Stealing a pet is an awful crime which can cause families great emotional distress whilst callous criminals line their pockets."
She said changing the law would acknowledge that "animals are far more than just property and will give police an additional tool to bring these sickening individuals to justice".
The Pet Abduction Bill is being brought forward by Conservative MP Anna Firth.
The Southend West MP told the OceanNewsUK low prosecution rates made stealing a pet a "low-risk, high-reward crime".
"I just find it unbelievable that we treat the loss of a living creature, a member of our family, as if it is a power tool or a laptop."
Toni Clarke, who lives in rural Norfolk, lost her Siamese cat, Clooney, in 2013. He was rarely let out and she feared he might have been shot. But tracker dogs found no trace of him.
Ms Clarke reported her missing cat to the police, but found them uninterested. In 2018, she discovered his microchip details had been scanned by two different vets, neither of whom contacted her.
She told the OceanNewsUK she had never stopped searching for Clooney, who would now be aged 14, all over the country, or campaigning for cat thefts to be taken more seriously.
'Traumatising'
Her group, Pet Theft Awareness, has lobbied for a change in the law to get pets recognised as "valued living possessions" rather than inanimate objects. and for jail sentences for pet theft.
It has also called for compulsory scanning to increase the chances of stolen and missing pets being reunited with their owners.
A report by the group found that, in 2021, police-recorded cat theft increased by 40% on the previous year and had more than quadrupled since 2015.
Ms Clarke says the real figures are likely to be much higher, as police forces' willingness to record the crime varies dramatically across the country.
She says the impact on her of losing Clooney has been "profound and traumatising", not least the "open-ended grief of not knowing what happened".
Annabel Berdy, senior advocacy officer at charity Cats Protection, agrees it is important to extend any new criminal offence to cover cats.
She told the OceanNewsUK: "If you included dogs from the outset without cats, given that those are the two companion animals, it might drive exploitative criminals or people looking to steal animals for money towards cats."
The law should also recognise "the very similar emotional value and attachment that owners will have with their cats, as they do with dogs", she added.
Last May, Environment Minister Mark Spencer promised that measures in the Kept Animals Bill, including those covering pet abductions, would be enacted by other means, including using private members' bills.
Ministers are now understood to be "very sympathetic" to the bill being put forward by Ms Firth.
It is much harder for a private members' bill to become law but its chances are increased if it secures government support.
To become law, the bill would have to pass all its usual stages in the Commons and Lords before a general election was called.
Ms Clarke told the OceanNewsUK that, with nearly half of households owning a pet, it would be a hugely popular measure.
She said that while research had shown owning a pet was good for people's mental well-being, having them stolen caused "trauma and anguish", and new laws were needed to protect pets and their owners.
