Santner, who took 13 wickets in his last Test in India as the Blacks Caps won the second match on their way to a series sweep, was included in the squad only for the second and third Tests in Wellington and Hamilton.
"That's the likely big decision to make," Stead told reporters on Monday.
"I guess Mitch's selection is on the back of what happened in Wellington last (season) with it being very, very spin-friendly. But we will monitor that when we get up there and have a look."
In March, Australia offspinner Nathan Lyon took 10 wickets at Basin Reserve as New Zealand, who did not play a specialist spinner, fell to a 172-run defeat.
The Black Caps fielded four seamers in the first clash against England in Christchurch and Nathan Smith, who made his Test debut, looks the most likely to make way if Santner comes into the side in Wellington, where the series resumes on Friday.
While a flurry of dropped catches were the most notable feature of New Zealand's performance in the first Test, some of the home batters, including opener Devon Conway and wicketkeeper Tom Blundell, continued poor runs of form.
England's Chris Woakes (left) took the wicket of the out-of-form Tom Blundell in the first Test. (AP PHOTO)
Stead said there would be no changes to the squad before the Wellington match and shrugged off the suggestion that Glenn Phillips, who is in fine form, might be moved above Blundell in the batting order.
"We've got a lot of faith in Tom Blundell," he said. "Glenn Phillips has done an amazing job batting with the tail as well, and has become really adept at that.
"Tom Blundell played some critical innings for us ... and we think he's our man to keep coming right, and we have a lot of faith that he will do so."
Conway has scored just 53 runs in his past five innings, but Stead backed him to come good too.
"Everyone wants more all the time," Stead said.
"I think opening in New Zealand conditions can be some of the toughest conditions in the world to play.
"Devon's a class player, and you don't lose that class over a week, a month. He wants more, don't get me wrong, but we still have a lot of faith in the way that Dev plays."
