A rescue team of helicopters, divers and trackers combed the area while friends gave out missing posters in villages and to fishermen along the river.
Mr Van Zyl's footprints were later spotted leading to the river bank and trackers found his backpack nearby.
Sakkie Louwrens, who was part of the search team, said police suspected two Nile crocodiles may have eaten Mr Van Zyl.
'We found what could possibly be human remains in them,' he told The Telegraph.
Police and animal nature conservation services decided to shoot the reptiles.
The remains are being tested by forensic experts to see whether they belong to Mr Van Zyl.
At least four people have been killed by crocodiles in Zimbabwe in the past month.
In March, villagers cut open a crocodile and found the remains of an eight-year-old boy inside the beast.
The shocking scene was captured by an eyewitness with a smartphone in the village of Mushumbi Pools in northern Zimbabwe's Mashonaland Central Province.
Villagers suspected the crocodile had killed and eaten the young boy, and shot the animal dead, the Daily Mail reported.
Zimbabwe has recently been hit by heavy rain, raising river and dam levels, which can bring crocodiles to areas where they are not normally seen.
A crocodile was recently shot dead in Beatrice, a farming community in the neighbouring province of Mashonaland East, with what were believed to be the remains of a fisherman in its stomach.
In November, last year a 13 year old boy who was fishing to pay for his school fees was killed by a crocodile in southern Zimbabwe.
Owen Chianga and his friend, Liberty Ruzivo, 15, were attacked by two crocodiles while they were fishing in the Save River near the village of Birchenough Bridge.
Nile crocodiles typically feed on fish, antelope and zebra, which they snatch from the shallows and before engaging in a twirling, drowning method known as 'the death roll'.
