The Bangladesh Cricket Board are considering the offer for the game scheduled for November 22-26
India want next month's second Test against Bangladesh at Eden Gardens to be their first day-night affair in the five-day format.
The World Test Championship leaders have proposed that the match on November 22-26 is played with the pink ball.
Neither side have ever played a Test under such conditions. Since its inception in 2015, only 11 day-night games have been played in the longer format, with Australia hosting five of them.
New BCCI chairman Sourav Ganguly, who officially began his new role this week, is a supporter of the innovation while captain Virat Kohli is said to be "agreeable" to the idea.
"They [BCCI] have proposed [Day-Night Test] us and we will let them know after thinking a while," Bangladesh Cricket Board operation chairman Akram Khan told reporters.
"We have received the letter two or three days and we will take a decision in this regard but we have not discussed about it. We will let them know about our decision within one or two days."
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Both India and Bangladesh have previously turned down offers to play a day-night Test and it remains to be seen whether their stance has softened.
BCB chief executive Nizamuddin Chowdhury added: "At first we have to take the consent of the players and members of the team management about it.
"It is totally a technical matter [playing day-night Test from skill perspective] and we have to consider that there is preparation required to play a Test with a pink ball."
The Test series kicks off in Indore on November 14 in what is Bangladesh's first match in the WTC. India have already played five times in the current cycle and have a healthy 240 points.
The Tigers' tour kicks off with three T20s, starting on November 3 in Delhi.