The batsman has started five days of isolation after taking his test in London on Wednesday evening.
All-rounder Michael Bracewell and two members of the backroom staff, Vijay Vallabh and Chris Donaldson, have also tested positive.
Conway and Bracewell should both be available to play in the final Test at Headingley, which starts on 23 June.
“The rest of the touring party have returned negative results and will continue following the tour health protocols of symptom reporting and subsequent testing if required,” a New Zealand statement said on Thursday. “No replacement players are being sought at this stage.”
New Zealand, the world Test champions, suffered a second successive five-wicket defeat by England at Trent Bridge on Tuesday, which gave the hosts an unassailable 2-0 lead in the three-match series.
Captain Kane Williamson missed the match in Nottingham after testing positive for Covid.
New Zealand have also lost pace bowler Kyle Jamieson, who suffered a lower back injury during the defeat in Nottingham.
© Agence France-Presse
