The tourists must bat out the final day of the fifth Test in Chennai on Tuesday to avoid a 4-0 defeat after India amassed 759-7 declared – their highest ever Test score and the biggest England have conceded.
Karun Nair, who made just 17 runs in his previous two Tests, became only the second India batsman to hit a triple century before England began their reply by reaching 12-0 at stumps.
Cook faces a period of soul-searching regardless of the outcome, as he weighs up relinquishing the burden of captaincy against spearheading another Ashes tour Down Under next winter.
He intends to discuss the dilemma with director of cricket Andrew Strauss in the coming weeks and Bayliss, who flew home to Australia on Monday night for a hernia operation, has no plans to intervene.
“He has done it for so long, whether I say yes or no, it won’t change his mind,” said Bayliss.
“If he keeps going – fantastic. If he stops, that is up to him, and I have got no problems with that either.
“I won’t actually sit down to have a chat with him. We’ve got six or seven months until our next Test match, so there is plenty of time for him to have a think about it.
“He has been around long enough and knows his own mind to make the right decision. I think he deserves everyone letting him do that.”
Nair’s hesitancy when resuming on 71 of India’s 391-4 gave little hint of what was to come: a knock of 303 not out, the third century of which took him just 75 balls.
England missed chances to stop him – Joe Root dropping a catch when he was on 217 and a stumping opportunity for Jonny Bairstow when on 246 – but Nair survived to compound the tourists’ misery.
“That’s the best knock of my life,” said the 25-year-old
“The first hundred is always important and when I got that I didn’t feel any pressure and just played my shots.
“The pitch is getting worse and for a new batsmen it won’t be easy to play. Hopefully it will turn more tomorrow.”