Komphela, who lost the national coaching race to Gordon Igesund, is poised to replace Serame Letsoaka, whose contract expired last month.
His representative, Tim Sukazi, met with SAFA CEO Robin Petersen on Tuesday, where the footballing body officially offered Komphela, an ex-Bafana skipper, the role.
Sukazi said in the Cape Times: "We are still in the early stages of our discussions with the national association but we want to conclude these dealings quickly so that Steve can focus.
"We will be doing whatever is necessary to try and have everything sorted in the next 24 hours.
"Should all things be equal, we'd like to take the job. It is a national office, and Steve is a firm believer that when the nations calls we oblige."
He added: "It is only fair to Stars that that the whole process is concluded swiftly so that, should Steve be moving, they can do whatever they can to replace him."