The dog, who would be 117 in human years, was rescued by Ray and his wife Mary 17 years ago from a couple who could not care for it.
The previous holder of the oldest dog title is believed to be a 25-year-old Jack Russell called Meg from Somerset, that died two years ago.
Mr Bunn said: "Jack was a part of our family, and we just can't believe he has gone. He was the oldest dog in the country and for him to die in this way is a tragedy."
His wife added: "We won't be getting another dog. Jack was irreplaceable. He had his own unique character and was loved by so many people. He was very well known in the area for being the country's oldest dog. Everybody will be so upset.
"Jack had me up at 5.45am and I shouted at him. I feel guilty about that now, but Jack knew he was loved and cared for. He had a very long and happy life, and I take comfort from that."
Mr Bunn said the incident had been reported to police in Hartlepool.