They said the Wagga Wagga man handed himself in about 11am Monday, and would be charged with one count of destroying or damaging property. He has been granted bail to appear in court in September.

Earlier, police appealed to the public for help to identify two people seen near the Dog on the Tuckerbox statue in Gundagai when it was vandalised on the weekend.

Residents and visitors found the statue knocked over on Sunday morning. The dog and the tuckerbox had fallen into the surrounding wishing well.

NSW Police on Monday released images of a man and a woman spotted near the statue at the time of the incident.

A motorist on the side of the Hume Highway noticed a man vandalising the statue about 4pm on Saturday.

The man is described as being of medium build with black hair. He is depicted in the image wearing a black jumper, light tracksuit pants and black glasses.

The woman is described as being of thin build with dark hair. She is pictured wearing blue jeans, a black top and brown jacket.

For more than 80 years the bronze dog on the tuckerbox has been sitting obediently as tourists visit the Australian icon.

The site's current leaseholder Denny Allnutt said on Sunday the low act was not welcome in the community.

"I'm absolutely gutted and really devastated; it's so tragic to see the dog being the target of such a nasty sense of vandalism," she said.

"Visitors are shocked and saddened and we would've had about 100 people come today who are disappointed to find it like this, especially those who have traveled a long way."

Ms Allnutt's lease ends on Wednesday and said this tragedy could impact the new leaseholder's future plans.

"This will hugely impact her as we don't know at this stage if the dog will be repaired by the time the site opens again on August 10," she said.

"The dog's ears have broken off and the historic old wishing well, made of fragile sandstone, has been damaged; the dog can be repaired, but the well can't be.

"The new leaseholder has got great plans for the side and it was all on track, but suddenly there is potentially no dog."

The dog was famously stolen in 1981 by a group of Canberra university students as part of a prank.

"The dog was well looked after and then handed into Collingullie Police Station, but this was nothing like that," Ms Allnutt said.

"I'm sure the dog will be back on his tuckerbox and will remain there for years to come."

The Cootamundra Gundagai Council is working with the Riverina Police District to find those responsible.

Anyone with information or dashcam footage that may be able to assist investigators is urged to contact Gundagai Police Station on 02 6944 0499 or Crime Stoppers on 1800 333 000.