Cairns, 51, was on life support after having cardiovascular surgery in Australia's capital city.
His family have said he has had further heart surgery at St Vincent's Hospital in Sydney.
A statement from his wife Melanie also thanked "everyone for their warm wishes, prayers and kind words".
Considered one of the best all-rounders of his generation, Cairns played 62 Tests, 215 one-day internationals and two Twenty20 matches for New Zealand between 1989 and 2006 before becoming a television pundit.
