The English-language Arab News did not name the prince or his victim, but said a senior member of the family and government, Crown Prince Salman, had "cleared the way for the possible execution of a prince convicted of murdering a Saudi citizen".
In a message about the case to Interior Minister Prince Mohammed bin Nayef, Prince Salman said: "Sharia (Islamic law) shall be applied to all without exception", it was reported.
Prince Salman's message followed a statement from the victim's father that he was not ready to pardon the killer and he was not happy with the amount offered as blood money.
The families of murder victims are encouraged by authorities to accept blood money instead of insisting on execution.
The paper quoted Crown Prince Salman's message as saying: "There is no difference between big and small, rich and poor ... Nobody is allowed to interfere with the judiciary's decision. This is the tradition of this state. We are committed to following the sharia."
An Interior Ministry spokesman was not available for comment.
The Arab News is part of a media group chaired by a son of Crown Prince Salman, who is deputy prime minister and minister of defence.
The kingdom, which follows a strict version of sharia has been criticised in the West for its high number of executions, inconsistencies in the application of the law, and its use of public beheading.
Saudi Arabia had executed at least 47 people as of May 2013, according to Amnesty International.
Members of the ruling family are rarely known to be executed. One of the most prominent cases was Faisal bin Musaid al Saud, who assassinated his uncle, King Faisal, in 1975.
The family is estimated to number several thousand.