Earl Spencer, the brother of the late Princess Diana, says local news reports on his divorce case are harming his children and he will ask South African courts to put a stop to them.
South African law forbids newspapers from reporting details of divorce cases.
But some have carried allegations by lawyers for Spencer's estranged wife, ex-model Victoria Lockwood, that he had up to a dozen affairs.
The hearing is being held to decide whether the divorce case will be held in Britain or South Africa.
Earl Spencer is denying allegations that he committed adultery with 12 women within months of marrying a former model and says the reports are harming his children.
The allegations emerged at the start of a hearing to decide whether Spencer's divorce from his estranged wife, Victoria Lockwood, will take place in Britain or his adopted home of South Africa.
Cameramen and photographers thronged the street outside the Supreme Court as Lockwood and then Spencer arrived for Wednesday's hearing at Court 17 - an indication of the high media-profile the case is receiving.
Spencer's lawyers informed the Cape Times newspaper on Tuesday night that he will apply to Cape Town's Supreme Court to prevent it publishing further details.
The application will be heard later on Wednesday.
In his legal notice to the Cape Times, which printed excerpts on its front page, Spencer said the reporting was affecting his children, especially his eldest daughter Kitty, 6, who attends school in Cape Town.
Earl Spencer says that his daughter's friends were talking about the case and she had seen newspaper posters on her way to school that carried details about his alleged affairs.
The Cape Times ran its front page story on Wednesday on the legal move, but said it would not be printing any further details from the divorce case.
"On Tuesday evening our lawyers were informed that the Earl was going to seek to restrain the Cape Times from publishing any information on the divorce proceedings currently under way. Some of the reasons, the Earl claimed that we had breached and contravened section 12 of the divorce act. The act says in terms... we can only record the identities of the of parties."
SUPER CAPTION: Ronald Morris, Cape Times High Court Reporter
There were also reports that the Earl's lawyers wished to restrict TV coverage too.
"I understand, although I haven't had official confirmation, that the Earl's lawyers have also approached the South African Broadcasting Corporation and asked them to desist from filming the Earl on his way to and from court. That to me would seem to be somewhat extraordinary, because it is a case which has affected wide public interest and we would be filming the Earl in public."
SUPER CAPTION: Ed Herbst, Correspondent for SABC
Foreign media are not bound by the law and the story of Spencer's alleged affairs has made headlines around the world, especially in Britain.
The Cape Times, which first started breaking the law on Tuesday morning but was quickly followed by other local media, said that it was campaigning for press freedom.
Spencer wants the divorce case to be settled in his adopted home of South Africa, while Lockwood wants it to be held in Britain.
She is arguing that the case involves two British nationals and most of the earl's assets are in Britain.
If the case is heard there Lockwood would likely get a larger court-ordered settlement than in South Africa.
The couple married in September 1989 and now live in separate houses a few streets from each other in Cape Town's plush suburbs. They have four children.