Batting duo Kane Williamson and Henry Nicholls were both unable to train on Thursday and left training early on Wednesday due to illness, ahead of the Third Test which starts in Sydney on Friday.
South Africa-born 23-year-old Phillips has flown across the Tasman to link up with the squad on Thursday.
The right-handed batsman is uncapped at Test level but has represented the Black Caps 11 times in the T20 format.
Spinner Mitchell Santner is also understood to have fallen ill and his availability is also in doubt.
New Zealand coach Gary Stead was full of praise for Phillips who scored a century for New Zealand A against England in November.
“Glenn has been in terrific form this season and has good versatility with where he can bat and the roles he can play,” Stead said.
“He’s been around the Black Caps T20 team for the past couple of years so should feel comfortable in the environment.
“We’re still hopeful Henry and Kane will pull through and we will give them every chance to prove their fitness.
“If one or both are ruled out then we will consider our options and most importantly how they balance the side for this final Test.”
New Zealand opener Tom Latham stood in on the media duties for the unwell Williamson and backed the skipper to be available.
“He’s a great leader,” Latham said about Williamson.
“If there’s any chance of him playing, even if it’s a small chance, he’ll certainly be playing.”
Williamson has only managed 57 runs from four innings during the Test series against Australia.
The Black Caps have already lost debutant Lockie Ferguson and Trent Boult for the series due to injury, having lost both Tests by a considerable margin.