Judges of the Court of Appeal, High Court and advocates in the academia, civil societies, corporations, and prosecutors can apply for the job should they fulfil the set out qualifications.
Justice Mwilu and Justice Njoki Ndung’u are the only women at the Supreme Court.
Names of top judges and senior lawyers jostling to succeed Mr David Maraga as Chief Justice are being floated as the Judicial Service Commission next month starts the process to recruit his successor in what is billed as a high stakes race.
Among top contenders is former Attorney-General Githu Muigai, Supreme Court judges Njoki Ndung'u, Isaac Lenaola, Mohammed Ibrahim, Smokin Wanjala and the deputy chief justice Philomena Mwilu.