The Independent said a second image showed the couple in front of another dead cat.
The image was initially posted on the website of Legelela Safaris, which specialises in big-game hunting trips in South Africa and neighbouring countries.
The website was offline on Wednesday.
TimesLIVE contacted Legelela Safaris on Wednesday to establish if it was aware of reports about the lion hunt, but was informed there would be "no comment at all".
The images have subsequently been widely shared on social media, drawing outrage from some quarters.
The founder of Campaign to Ban Trophy Hunting, Eduardo Goncalves, told The Independent that the lion may have been shot while trapped in an enclosure.
The Mirror reported that the couple owned a taxidermy business and described themselves as "passionate conservationists".
When approached for comment on the matter, they told the Mirror: "We aren’t interested in commenting on that at all. It’s too political.”