A police spokeswoman told the Herald on Sunday they were "following up" Spicer's complaint.
Barnhill said she had permission "in writing" to keep Cogs, but could not show it to the Herald on Sunday.
She said it was with police and she would not comment further while the investigation was under way.
It is understood she has renamed the cat Missy Tiggs.
Spicer insisted the arrangement was only supposed to be temporary. She was prepared to "go all the way to court" to get Cogs back.
The curious case is the second cat fight in a week.
The Herald on Sunday last weekend reported the online catfight between teen social media sensation Caitlin Davidson and broken-hearted cat owner Joyce Quah over a purebred chinchilla persian cat.
Davidson posted pictures of a cat she claimed was a rescued stray to the almost half a million followers of her Caito Potatoe social media pages. Quah believes the cat is her missing pet, Chloe.
Both parties have gone to police over the dispute and have been told it is a civil matter.
Davidson last week referred comment to her father, Tom, who said his daughter had been subject to online abuse following Quah's claims. He did not want to comment this week.
Quah said on Friday she had filed an application with the Disputes Tribunal asking for a DNA test to prove the cat's identity and for future ownership to be based on the finding.