Alex Hales, 27, filled the slot in the summer series against Pakistan but averaged just 18 in four Tests.
Asked if England could be about to pick a new opening partner for captain Alastair Cook, Bayliss said: "Possibly.
"If a player's good enough, he's old enough. I wouldn't have any problem [picking him]."
If selected, Lancashire's Hameed would be the ninth batsman to fill the role alongside Cook since Andrew Strauss retired in 2012.
Hameed has scored 1,129 runs in the First Division of the County Championship so far this season, including four centuries, at an average of 54.
It is just 12 months since the then England Under-19 captain told how he had been getting tips from potential partner Cook ahead of the youth Ashes Test against Australia.
Bayliss added: "There have been some young players making some runs who are coming into calculations - which is very exciting."
England are also monitoring Northants batsman Ben Duckett ahead of the tour to Bangladesh, said his county's director of cricket David Ripley.
Another point for discussion for England's selectors will be whether to pick Liam Dawson as the third spinner.
The Hampshire left-armer conceded two for 70 in eight overs on his one-day international debut against Pakistan on Sunday.
"There are one or two positions in a squad of 16 that could go either way," added Bayliss.
"He will certainly be in discussions - he's a guy who's got something about him.
"He bats, bowls and he's a pretty good fielder as well. But it's his game sense [that's impressive] - he knows how to play the game, almost knows what's going to happen before it happens."
Meanwhile, the England and Wales Cricket Board (ECB) has advised Yorkshire that Joe Root and Jonny Bairstow should be rested for their remaining two County fixtures.
Yorkshire, who are second as they go for a third County title on the trot, have games coming up against Somerset and leaders Middlesex.
Warwickshire's Chris Woakes and Surrey's Jason Roy are available for the Royal London One-Day Cup final between their sides on 17 September but the ECB has asked for Woakes to be rested from their last county fixture against Lancashire.
A decision on other players will be made after the Twenty20 international against Pakistan on Wednesday.
England coach Trevor Bayliss says he would have no problem selecting Haseeb Hameed, 19, as a Test opener for the winter tours to Bangladesh and India.