The other four names on the list are judges Selby Baqwa, Lebotsang Bosielo, Brian Spilg and advocate Mbuyiseli Madlanga, Beeld reported on Wednesday.
The Judicial Services Commission (JSC) will interview the candidates on February 24, then forward four of the candidates' names to President Jacob Zuma.
According to the JSC's spokesperson, CP Fourie, the JSC will not name its preferred candidate and Zuma will have to decide which of the candidates he wants to appoint.
Gauntlett made headlines in 2012 when he was ignored as a potential candidate for a judicial appointment in Cape Town.
After a public outcry, the JSC eventually acknowledged that Gauntlett had not been appointed because he "was short-tempered" and "did not have the humble qualities expected of a member of the judiciary".
An even bigger outcry followed, after which more than 60 of the country's leading academics and institutions declared their support for Gauntlett. – Sapa.