Guptill said today he was feeling noticable improvement and is optimistic of a speedy return.
"It's feeling a lot better than it was last week. I've been doing some good work with the physio here and getting to being right for the game next week," he said.
Guptill, the last of whose 31 tests was also in England, at Leeds in May 2013 is out for the tour opener, a four-day first-class fixture against Somerset at Taunton starting on Friday, which leaves New Zealand's resources thin.
Several players are still involved in the Indian Premier League. However three players were added to the squad, partly for experience, and partly to act as cover, including seamer Ben Wheeler and Jacob Duffy, while new limited-overs selection, allrounder Mitchell Santner, is also with the squad.