Both players have had tiring weeks at the ATP Cup, with Djokovic leading Serbia into the final of the competition while the Russian team reached the semifinals.
Djokovic played back-to-back three-setters in the quarterfinals and semifinals of the ATP Cup. The Serb's initial decision to enter Adelaide had been surprising because he does not usually play the week before the Australian Open, and had a busy run-up to the tournament that also included an exhibition event in Abu Dhabi.
"We understand his decision," said Adelaide's TD Alistair McDonald in a statement.
The world No. 2 would have been the top seed at Adelaide.
Medvedev had noted he was fatigued from having to go from playing in Perth to Sydney during the ATP Cup.
"But with the load he's had this week and the travel issues he's had from Perth it's been a bridge too far," said Auckland TD Karl Budge, speaking to New Zealand press. "Unfortunately they were drawn in Perth, which is the hardest travel schedule."
Medvedev's fellow Russian Karen Khachanov has also been playing ATP Cup but is still entered in Auckland.
Medvedev would have been the top seed at Auckland.