The Swans have shortened their list of targets, with Sheffield United player-coach Gary Speed, Exeter manager Paul Tisdale, Aidy Boothroyd and Gus Poyet all rumoured to be in line for the job after Martinez's defection to Wigan.
After a turbulent fortnight, including the departures of Martinez and support staff Graeme Jones and Kevin Rees, the club are looking for stability.
Chairman Huw Jenkins told BBC Sport: "We are down to about six main people. I know in my mind who we're looking at but I'm not going to give names.
"Hopefully it will push on between now and the weekend."