Just under a year ago their other cat, Oreo, lost an eye after a similar attack.
The latest outrage happened on Tuesday.
Animals charity the SSPCA fears a sick repeat offender is on the loose, and is appealing for the public for information.
Rhea is fighting for her life as vets perform emergency surgery to remove part of her bowel, which was penetrated by the gun pellet.
Owner Alan Smith, of Eriskay Court, said: “Rhea is touch and go at the moment, and there is the risk of infection.
“The vets are concerned the pellet went through to her bowel. Because of the nature of the wound, the vet thinks she was shot at close range”.
He added: “This is the second time this has happened to us and it’s hard not to think we are being targeted. It’s devastating.
“She is a lovely cat and wouldn’t bother anyone. She doesn’t deserve to be going through this.”
It is the third time a cat is known to have been shot in Falkirk in less than a year.
Scottish SPCA inspector Andrew Gray, appealing for information, said: “We are incredibly concerned for Rhea and cannot imagine what the Smiths are going through.
“Rhea is still receiving emergency treatment and we hope she pulls through.
“This is the second time the family have gone through this, and the third cat within a two-mile radius since last June to have suffered injured from an airgun.
“We are concerned this person or persons is going to commit more crimes against animals.
“This was a deliberate and incredibly cruel attempt to wound, or kill, Rhea. The person must be caught before they do this again.
“We are working closely with Police Scotland due to the nature of this criminal act.
“Anyone with information should contact our confidential animal helpline on 03000 999 999 or Police Scotland on 101.
“Owning an air rifle without a licence has been illegal since December 2016, this is a move we welcomed due to incidents such as this.
In June last year the injury suffered by the Smiths’ other cat was described by SSPCA inspector Robyn Gray as “deliberate, calculated, and entirely cruel.
“Poor Oreo was targeted on two separate occasions. In the first instance the wound looked like a scratch and the pellet wasn’t visible, it was only when Oreo received the injury to his eye that it became clear this was a deliberate attack”.
At that time the SSPCA warned Falkirk pet owners to be vigilant and aware of the threat.
After that attack, Mr Smith said: “I’d urge anyone with information to please contact the Scottish SPCA no matter how relevant you think it may be, it could help catch the cruel individual who did this to Oreo.”
Now he and his wife face the trauma of another badly wounded animal, amid fears the person responsible could strike again.