Cummins and Starc sat out Australia's 3-0 series win in Zimbabwe over the last week but will play in the series in South Africa, which starts in Durban on Thursday.
Edwards, 26, is in line to make his ODI debut after diverting from the Australia A tour of India, while fast bowlers Billy Stanlake and Spencer Johnson return home after competing in Zimbabwe.
Stanlake suffered a side strain and will have medical scans on his return to Australia. Johnson was deemed surplus to requirements after playing two of the three matches in Harare.
Edwards has played a single Twenty20 international against Pakistan in January and offers cover with Cameron Green struggling with a sore shoulder, which caused him to sit out the three fixtures in Zimbabwe.