Called "confusing" by India captain Virat Kohli, the Umpire's Call has been a subject of controversy for a while now.
Ashwin became the fourth Indian to take 400 Test wickets when he dismissed Jofra Archer on the second day of the day-night third Test.
Spin legend Anil Kumble is the leading wicket-taker in India with 350 scalps at 24.88.
Bumrah, who missed the final Test of the Australian tour, is the favourite to share the new ball along with veteran pacer Ishant Sharma in the Test series against England which begins at Chennai on Friday.
Glenn Maxwell and Sheldon Cottrell are likely to be released ahead of the 2021 edition, according to a team source.
The camaraderie between Kumble and Rahul, with both coming from Bengaluru, should also help the team.
The 90-year-old was also a former cricketer and BCCI administrator.
Kumble resigned from the high-profile role in 2017 after his relationship with skipper Virat Kohli became 'untenable'.
The committee, chaired by former Indian spinner Anil Kumble, unanimously agreed to recommend a ban on saliva although players will still be allowed to use their own sweat to try and achieve the fabled 'reverse swing'.
Gambhir acknowledged the feats of other great Indian skippers like Ganguly and Dhoni but placed Kumble right at the top.