The 54-year-old played 67 Tests from 1992 until 2006, and was a key member of Australia's all-conquering side under the captaincy of Steve Waugh.
Martyn retired in 2006, finishing his Test career with 4406 runs at an average of 46.37 that included 13 centuries.
The middle-order batter was also a key part of the Australia side that won the 2003 ODI World Cup in South Africa.
Speaking on behalf of Martyn's family, close friend and former teammate Adam Gilchrist said: "He is getting the best of treatment and (Martyn's partner) Amanda and his family know that a lot of people are sending their prayers and best wishes."
Cricket Australia CEO Todd Greenberg added: "I'm saddened to hear of Damien's illness. The best wishes of everyone at CA and in the wider cricket community are with him at this time."
Former teammates and opponents have taken to social media to show their support to Martyn and his family.
The International Cricket Council has rated the pitch used for the Boxing Day Ashes Test at the Melbourne Cricket Ground as "unsatisfactory", issuing the venue with one demerit point.
Usman Khawaja is all but certain take his place in Australia's XI for the fifth and final Ashes Test in Sydney, but the batter's international future remains unclear.
England ensured the Ashes would not end in a clean sweep by securing a four-wicket win over Australia after just two days in the fourth Test at Melbourne.
England were skittled for 110 inside 30 overs on the opening day of the Boxing Day Ashes Test in Melbourne, after earlier bowling Australia out for 152 thanks to Josh Tongue's five-wicket haul.
