Already facing a 3-0 deficit, England had hoped to end the best-of-five series on a positive note but despite hitting 477 in their first innings, Alastair Cook's side face the possibility of suffering an innings defeat in Chennai.
India began the fourth day on 391-4 and although it did not take long for Liam Dawson to claim his first Test dismissal with the wicket of Murali Vijay (29), Nair soon took centre stage after previously only recording a highest score of 13 in his previous two Test appearances.
Ravichandran Ashwin (67) and Ravindra Jadeja (51) both provided support down the other end but it was the performance of Nair that captured everyone's imagination as he first scored the highest ever total for an Indian batsman against England.
The 25-year-old would soon become just the second Indian player to reach a triple ton and his innings only came to an end at 303 not out when India finally declared on 759-7.
Moeen Ali and Adil Rashid contributed combined figures of 2-343 off 70.4 overs as England were brutally exposed by the free hitting of Nair and his teammates and it left England needing 282 runs to avoid an innings defeat.
Keaton Jennings (9*) and Cook (3*) ensured that England reached the close on 12-0 but they will need a strong start on Tuesday to give the team some sort of platform to build on as they look to claim a draw.