The head coach is facing significant scrutiny and is at risk of losing his job, as his team has yet to secure a win in the Premier League.
The Saints were 2-0 up against Leicester City, but lost 3-2, which only compounded the pressure on Martin.
Per The Express, there are five major candidates for the job, but it is not clear if they are all interested.
Graham Potter, formerly of Chelsea and Brighton, is seen as an ideal candidate for the young squad.
Former England boss Gareth Southgate, ex-Leeds boss Jesse Marsch, former FK Bodo/Glimt manager Kjetil Knutsen, and Marcelo Gallardo are the other candidates.