Seam bowler Blair Tickner, batter Henry Nicholls and bowling coach Shane Jurgensen all face five days of isolation after returning positive rapid antigen tests on Friday.
Black Caps boss Gary Stead said: "We're not sure how it happened. The three guys who have caught it haven't been socialising together or anything so we're not sure what the source is.
"They're all feeling okay but they've been struck down by that virus. All we can try and do is minimise the risk out there.
"We're not in a situation where we are in a lockdown bubble or anything, but we're not going to big gatherings and if we're inside we're making sure we're wearing masks.
"Everyone is triple vaccinated and we'll be testing everyone in the party for the next three days. Anyone who has symptoms goes through the normal protocols of reporting to our doctor."
While the rest of the squad was given the all clear, their hopes of getting out on the field were scuppered by rain. A saturated outfield ended any hopes of action on the opening day of the warm-up match between the Kiwis and Sussex.
New Zealand take on England in the opening Test match on June 2 at Lords. Two more Test matches will follow on June 10 and 23 at Trent Bridge and Headingley.
The tourists are currently without captain Kane Williamson, who is back home as he awaits the birth of his second child.
Tim Southee, Trent Boult and Daryl Mitchell are also away on Indian Premier League duties.