The question has been asked on numerous occasions.
The answer is clearly yes, as both are in action in the current game against India in Mumbai.
But, going forward, is it a realistic option for the England selectors?
The dynamics of this question have altered during the past year or so.
Previously, Buttler was considered the No1 wicketkeeper-batsman and Bairstow his understudy.
It was largely presented as an either-or choice.
Either you were for Lancashire’s Buttler behind the stumps, or you were for Yorkshire’s Bairstow, with precious little middle ground.
But as England’s team for next winter’s Ashes series starts to take shape, there would seem no reason why both cannot continue to play in the same Test XI.
Bairstow is the best wicketkeeper and has now made that position his own, fighting off a flurry of ill-informed criticism along the way.
He is also the best all-round batsman of the pair and ahead of Buttler in the either-or stakes.
But Buttler has done well too, facing up to the reality that Bairstow has gone past him and carving out a potential niche for himself at No7, the position from which he struck 76 yesterday to lead England to 400 before India ended day two of the fourth Test on 146-1.
Not often in their Test history have England been able to accommodate a specialist batsman at No7.
To do that normally requires the cushion of a top-class all-rounder, someone who can bat and bowl and thereby create space for an extra specialist batsman or bowler.
In Ben Stokes, England possess such an all-rounder, who provides them with an extra seam bowling option.
England, in fact, now have a wealth of players who can bat and bowl; Moeen Ali, Chris Woakes and Adil Rashid all have first-class hundreds and first-class five-wicket hauls to their name in addition to Stokes.
Bairstow and Buttler are all-rounders too, but Buttler’s route to a regular place is as a batsman.
As such, his performance in Mumbai will do him no harm, particularly in the way that he constructed his innings.
Everyone knows that Buttler can smash the ball out of the park for fun in one-day cricket, where he is as dynamic as any batsman in the world.
But his defence has been questioned at Test level and often found wanting, creating the impression that he is more of a one-trick pony than a player for all formats.
However, the Buttler on show at the Wankhede Stadium displayed an adaptability that could not be faulted.
When he came to the crease on the first evening, England were 249-5 having slipped from the comparative comfort of 230-2.
But he dug in with Stokes to lift the team to 288-5 at stumps, with both men reining in their natural attacking instincts.
Although Stokes fell early in the piece yesterday, Buttler advanced sensibly from his overnight 18, displaying a watchful attitude at first before opening out and playing his trademark strokes.
Praise came from many quarters – not least from Geoffrey Boycott, who had specifically called for England’s batsmen to stay in longer and wear down the bowlers.
Frustrated by top-order collapses that have become as routine as night following day, the former Yorkshire and England opener had demanded more application.
“The best batting today came from Buttler,” said Boycott on the radio.
“He showed a degree of common sense and maturity which we haven’t seen much of from him in Test cricket.
“He started sensibly, he had to be careful and wary, and the longer he was in he thought ‘I’ve got to look to score’ but he did it sensibly.
“Seventy-odd here was worth a hundred, easily, especially if England win the match.”
Boycott added that Buttler should be batting at No4 “because he’s that good”.
However, after Keaton Jennings’s hundred on debut on day one, the likelihood is that – barring injuries or disaster – England will enter the Test summer with a top-four of Alastair Cook, Haseeb Hameed, Keaton Jennings and Joe Root.
Buttler could slot in at No5 – or, indeed, at No4 should the opportunity arise – but Bairstow and Stokes make for a strong No5/No6, with No7 perhaps more of a natural position for Buttler, from where he will have more licence to counter-attack.
If England play two spinners, it would be difficult to accommodate Buttler with a full quota of seamers.
But England would surely need only one of Moeen Ali and Rashid to bowl spin in most English conditions, with back-up from Root if required.
A potential first team going forward, give or take conditions, could be: Cook, Hameed, Jennings, Root, Bairstow, Stokes, Buttler, Moeen at No 8 or Rashid at No 9, Woakes, Broad, Anderson.