Wek, 35, who revealed this during an interview with Sowetan, is in the country to shoot a commercial for Amarula, which is due to be flighted in South Africa and other countries.
Local fashion line Lunar, founded by fashion designer Karen Ter Morshuizen, dressed Wek for the advert.
Wek, who was born to a housewife mother and an education official father, said she was currently not dating and swore that her only present family was her dog. "It would be lovely to settle down and have kids, but I am too busy for that for now.
"My family have been asking me ... when am I going to have a family? And coming from a big family, my mother asks me about it all the time. So, I am working on that."
She said coming to South Africa brought back the memories of when she had returned to her native South Sudan to celebrate the new republic's independence.
"South Africa is wonderful in many ways than one. It is such a warm country, with warm people who are united despite coming from different tribes. I find a lot of similarities between the two countries in that way."
The former North African refugee who is now a British citizen says going back to her country to celebrate independence was one of the most exciting moments in her life.
"The warm welcome I received was amazing. They really embraced me as one of their own. It was very touching to see unity among the people of South Sudan."
Wek said even though most people have lived under extreme poverty, she was shocked to learn that many of them recognised her and knew what she was doing.