Australia are scheduled to play their first Test against Pakistan at Dubai starting from October 7.
Mitchell Starc says that Steve Smith's "leadership group" remark affected other reputations.
Hazlewood was ruled out of the touring party with back trouble.
Hazlewood snuffed out England's outside hopes of a come-from-behind victory when he dismissed captain Joe Root in the third over of the final day of the first-ever day-night Ashes Test.
Josh Hazlewood told Nathan Coulter-Nile to go after the English batsmen.
India opener Rohit Sharma took fifth spot while spinner Axar Patel clinched his career-best seventh rank on Monday
Josh Hazlewood suffered the strain while bowling the second over of the third day .
Along with the players, fixtures are also at risk including Australia's Test tour to Bangladesh from August, the one-day international series in India in September and the home Ashes series against England later this year.
How to Watch Australia vs Bangladesh Champions Trophy 2017 match. Read all about live coverage and ODI match action from Kennington Oval in London on June 5, 2017.
Offers made to Steve Smith, David Warner and others.
India will take on Australia in the crucial fourth and final Test match that will take place in Dharamsala from Saturday.
Umpire Chris Gaffaney gave Cheteshwar Pujara a scare when he almost raised his finger to a Josh Hazlewood delivery only to change his mind and scratch his hat.
The medium-pacer took care of Abhinav Mukund, Virat Kohli, Cheteshwar Pujara and the tail as India collapsed for the second time in the match.
Pujara was unbeaten on 79 at stumps, while Rahane was 40 not out, as India took a crucial 126-run lead over Australia at stumps of Day 3 of the 2nd Test
Australian all-rounder Mitchell Marsh termed the pace duo of Mitchell Starc and Josh Hazlewood as one of their biggest weapons going into the second Test match in Bengaluru.
Virat Kohli remained the highest-placed Indian batsman at second, while off-spinner Ravichandran Ashwin held on to his top position in both the bowlers' and all-rounders' list of the latest ICC Test rankings.
Australian skipper Steven Smith notched up his 21st fifty as the visitors finished the day with 143/4 and lead India by 298 runs at stumps on Day 2 of the first Test match in Pune.
Australia will take on an India 'A' side led by all-rounder Hardik Pandya in a three-day warm-up game at the Brabourne Stadium, starting Friday.