Bairstow is currently with the England T20 squad and played the first two T20 internationals of the five-match series before being left out of the third game. He will stay in New Zealand with the Test squad for the two warm-up games, while Denly tries to recover from the ankle ligament injury that ruled him out of the T20 series.
Denly is set to miss the first red-ball warm-up game which starts on Tuesday in Whangarei but is hopeful of being fit to play in the second three-day tour game that begins next Friday.
Bairstow was dropped from England's new-look Test squad after a disappointing Ashes series where he averaged 23.77 with a solitary half-century.
England's chief selector Ed Smith challenged Bairstow to come back from the axing as a stronger Test match player.
The majority of England's Test squad arrived in New Zealand this week and will travel to Whangarei on Monday with those that were involved in the T20 series following the 5th T20I at Eden Park on Sunday.
England Test squad: Joe Root (capt), Jofra Archer, Stuart Broad, Rory Burns, Jos Buttler, Zak Crawley, Sam Curran, Joe Denly, Jack Leach, Saqib Mahmood, Matthew Parkinson, Ollie Pope, Dominic Sibley, Ben Stokes, Chris Woakes, Jonny Bairstow (injury cover)