Buckingham Palace insisted that the Prince was simply trying to break up a fight between the two labradors and said that the pictures do not show definitively whether he actually struck them.
But animal rights groups condemned his behaviour as "sickening" and "cowardly".
The photographs show the Prince approaching the two dogs waving a long crook as they grapple over a dead pheasant.
One picture shows one of the animals cowering to one side as he swings the stick over his shoulder.
In another, it hovers above the spine of one of the dogs but is not shown making contact.
Onlookers said he appeared to take three swipes at one of the dogs while shooting in a field with his nephew Peter Phillips.
It is not illegal to hit a dog with a stick if it is to save it from some danger but the Prince could be prosecuted if it can be shown that he beat them unnecessarily.
Magistrates can impose fines of up to £20,000 or six months in prison for extreme cases of animal cruelty.
If found guilty, the Prince could also be banned from keeping animals and receive a criminal record.
A spokeswoman for the RSPCA said an investigation was not under way but declined to rule one out.
"We need to look at the photographs and my colleague the inspector will consider whether it is appropriate to take action," she said.
A Buckingham Palace spokesman added: "It has not been determined that he did strike the dog."
The Prince was criticised by animal rights groups three years ago for dispatching an injured pigeon with his walking stick.
Two years ago his father, the Duke of Edinburgh, was also at the centre of an animal cruelty row when a fox was clubbed to death with a flagpole during a shoot at Sandringham at which he was present.
And last year police investigated a claim that Prince Harry had killed a pair of rare hen harriers while shooting at Sandringham.
But no carcasses were found and the case was quickly dropped amid fears of a "dirty tricks" campaign.
The Princess Royal became the first member of the Royal Family to receive a criminal conviction in 2002 when she was fined £500 after her bull terrier Dotty attacked two children in Windsor Great Park.