Usman Khawaja will board the flight from Melbourne to Bengaluru on Thursday and could attend the team's training session on Friday, said the report.
The Pakistan-born batter has played 56 Tests, 40 ODIs and nine T20s for Australia. He was named Australia's Test Player of the Year on Monday.
India and Australia are all set to face each other in the upcoming four-Test Border-Gavaskar series, starting from February 9 in Nagpur.
Batter Usman Khawaja, who made his Test debut in 2011, has cemented his position in the Australian team over the years
Cummins captured three for 29 in a hostile 14 overs to set up the Australians for a dramatic final day charge for victory and a series clean sweep against the hapless Proteas
Australia gained an unassailable 2-0 lead over South Africa in three-match Test series, after claiming comfortable wins in the first two games
AUS vs SA, 3rd Test, Day 4 Live Updates: South Africa ended Day 4 of the ongoing third Test against Australia at 149/6 in Sydney. At stumps, Marco Jansen (10*) and Simon Harmer (6*) remained unbeaten at the crease as the visitors trail by326 runs
Usman Khawaja said it would be "harsh" if he was denied the chance of claiming his first Test double century by a declaration in Australia's third Test against South Africa.
Usman Khawaja produced some entertainment for the crowd as well after reaching his century as he broke into a happy jig.
Steve Smith 30th century: The video of Smith's "chainsaw" celebration after the century has gone viral and the former Australia captain explained the reason behind it during a press conference.