Finch aggravated his right calf injury during a practice session and is facing a race against time to be fit for the first ODI against India on September 17.
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 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.
Handscomb, who recently returned to Melbourne following the Test tour of Bangladesh, will depart for India on Saturday and will be available for selection in Sunday's series opener.
First Published: September 15, 2017, 11:37 AM IST