The Yorkshire wicketkeeper-batsman has been overlooked, with Jos Buttler selected to take the gloves for the two Tests starting in late November.
Here, the PA news agency compares the duo's statistics as Test keepers.
Ashes struggles cost Bairstow dear
Bairstow, 29, struggled to impress after being asked to perform different roles in England's batting order during this summer's drawn Ashes series.
After a pair of single-figure scores at Edgbaston he made 52 and 30 not out from number seven at Lord's before being promoted one place on his home ground of Headingley, scoring four and 36.
He made 17 at number seven and 25 at number six at Old Trafford but was bumped up to number five after Jason Roy's omission at the Oval, where he scored 22 and 14.
A series tally of 214 runs at an average of 23.77, with just that one half-century, makes grim reading – though Buttler, selected as a specialist batsman and beginning the series at number five, did not fare any better.
He too bounced up and down the order, batting at seven or eight in Leeds and Manchester, but improved after a slow start to the series.
He made only 247 runs at 24.70 in the series but finished off with scores of 41, 34, 70 and 47 to perhaps give him the momentum compared to Bairstow in the selectors' eyes.
Buttler must prove himself as Test keeper
Bairstow has generally been preferred to Buttler as England's five-day wicketkeeper, playing three times as many Tests with the gloves while the roles are generally reversed in white-ball cricket.
Bairstow's 48 Tests and 90 innings behind the stumps have brought him 169 catches and 12 stumpings for an average of 2.01 dismissals per innings.
That compares to 1.61 for Buttler, with 50 catches and no stumpings in 31 innings across 16 Tests.
James Anderson, injured for the New Zealand tour, accounts for the most catches by both keepers – followed by Stuart Broad, with 38 catches by Bairstow in 45 games and 14 in 16 by Buttler.
Bairstow also has 26 catches off Ben Stokes' bowling and 18 off Chris Woakes, as well as six and four respectively off the other touring seamers Jofra Archer and Sam Curran and a stumping off spinner Jack Leach.
Buttler has taken only two catches off Stokes in 13 Tests, while in three games keeping to Woakes and one with Curran he has not taken a catch behind off either man.
He has never kept to Archer or Leach in Tests, but will be familiar with the latter from their days together at Somerset and is a current Lancashire team-mate of new call-ups Saqib Mahmood and Matt Parkinson.
Also of note is the fact Buttler's Test batting average dips from 35.68 as a specialist batsman to 31.30 when keeping wicket, with no centuries as a designated keeper.
Bairstow, by contrast, averages 37.85 as a keeper – helped by five of his six career centuries – compared to 29.17 in his other 21 Tests.