"We have got to check with the medical guys in the next few days. They're suggesting he may not be able to keep but might be able to bat. We'll have to see how it responds," Bayliss told the British media.
England called up Hampshire batsman James Vince as cover on Thursday and wicketkeeper-batsman Jos Buttler could once again take over behind the stumps if Bairstow cannot recover in time.
Bayliss said Bairstow could also be asked to give up keeping duties completely after his injury has fully healed, to help him focus on his batting higher up the order.
"That'll be the hard thing, trying to convince Jonny," Bayliss added. "That will be a decision that is taken from the team point of view. He is a world-class batter, we know that, one of the difficult things might be convincing Jonny of that.
"If that was the way we went, it would certainly be a deep conversation. Jonny's a reasonable bloke... we know he wants to keep but there would be a lot of explaining and chatting."
India's victory at Trent Bridge saw them cut the hosts' lead to 2-1 in the five-match series.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by John O'Brien)
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