J Jayalalitha, chief minister of the state of Tamil Nadu - of which Chennai is the capital, died aged 68 on Monday.
The fifth Test is due to be held in the city from 16-20 December.
"We are keeping a close watch, and taking input from the local association about sentiments of the people," said board secretary Ajay Shirke.
India lead the five-match series 2-0 before the fourth Test in Mumbai, which starts on Thursday.
India's cricket board is considering whether to switch the final Test match against England from Chennai following the death of a local politician.