Finch got injured during practice at the M A Chidambaram Stadium and pulled out of the rest of the session, cricket.com.au reported.
In the event of Finch being ruled out of the first game, Australia may turn to Travis Head or Hilton Cartwright, the report said.
The Victorian had been rested from Australia's warm-up match on Tuesday due to the injury, which he suffered while playing for Surrey in England six weeks ago.
Uncapped all-rounder Cartwright opened in Finch's absence in the warm-up game, but was dismissed for a second-ball duck in Australia's 103-run win.
Australia's stand-in coach David Saker had indicated earlier that Head was the "preferred choice" for the No.4 slot in the line-up.
The left-hander, during an interaction with the media here this afternoon, said he would be happy to bat up the order.
"Hopefully I'll get the chance to go up the order again and bat at No.4," he said.
Aaron FinchFinch Injuryind vs aus 2017India vs AustraliaIndia vs Australia 2017M A Chidambaram Stadium
First Published: September 14, 2017, 5:33 PM IST