India trail the five-match series 2-1 after losing by 22 runs at Lord's last week.
"Nitish will fly back home and the team wishes him a speedy recovery," the Board of Control for Cricket in India said in a statement.
Reddy scored 45 runs and took three wickets in his two outings in the series.
The fourth Test starts on Wednesday at Old Trafford.
Left-arm quick Arshdeep Singh, who is yet to feature in the series, will miss the fourth Test after injuring his thumb during net practice.
Uncapped all-rounder Anshul Kamboj has been added to India's squad.