The Harlequins scrum-half, 27, jarred his shoulder after an accidental fall in training and will be given until Thursday night to prove his fitness.
England coach Stuart Lancaster will hand his place to Leicester's Ben Youngs if he does not recover in time.
"Danny is a concern. At the moment we're hoping he'll be fine for the game," Lancaster said.
Games since last defeat: 12 (38-21 by England, 1 Dec 2012) Games since last home defeat: 30 (32-29 by South Africa, 12 Sept 2009) Games since last home defeat by England: 5 (15-13, 14 Jun 2003) If Care is ruled out, Lee Dickson will be picked on the bench just three days after arriving in New Zealand and seven days after helping Northampton win the Premiership title.England captain Chris Robshaw says his team-mates are not intimidated at the prospect of facing the All Blacks in their own country.
"We've got to come out of the blocks all guns blazing and really start fast," Robshaw said.
"Eden Park will be a tough place to go, but we're looking forward to it."
New Zealand head coach Steve Hansen has recalled number eight Jerome Kaino to the side ahead of Kieran Read, who has been ruled out with concussion.
Kaino, 31, played more minutes than any other player during the 2011 World Cup but quit after the All Blacks' 8-7 final win over France to take up a two-year contract in Japan.
Recalled All Blacks number eight Jerome Kaino played more minutes than anyone else at the 2011 World Cup He returned to New Zealand this season and quickly regained his best form during the Super Rugby season with the Auckland-based Blues. He will play his 49th Test in an experienced back-row combination with captain Richie McCaw, who will play his 125th Test, and Liam Messam, who will play his 30th.
Aaron Cruden has been preferred at fly-half for the All Blacks despite an injury-hit season, while there are three uncapped players on the bench.
England: Mike Brown (Harlequins); Marland Yarde (Harlequins), Manu Tuilagi, (Leicester Tigers), Kyle Eastmond (Bath), Jonny May (Gloucester); Freddie Burns (Leicester Tigers), Danny Care (Harlequins); Joe Marler (Harlequins), Rob Webber (Bath), David Wilson (Bath), Joe Launchbury (London Wasps), Geoff Parling (Leicester Tigers), James Haskell (Wasps), Chris Robshaw (Harlequins, capt), Ben Morgan (Gloucester)
Replacements: Joe Gray (Harlequins), Matt Mullan (London Wasps), Henry Thomas (Sale Sharks), Dave Attwood (Bath), Tom Johnson (Exeter Chiefs), Ben Youngs (Leicester Tigers), Danny Cipriani (Sale Sharks), Chris Pennell (Worcester)
New Zealand: Israel Dagg; Ben Smith, Conrad Smith, Ma'a Nonu, Cory Jane, Aaron Cruden, Aaron Smith; Jerome Kaino, Richie McCaw (captain), Liam Messam, Sam Whitelock, Brodie Retallick, Owen Franks, Dane Coles, Tony Woodcock
Replacements: Keven Mealamu, Wyatt Crockett, Charlie Faumuina, Patrick Tuipulotu, Victor Vito, TJ Perenara, Beauden Barrett, Malakai Fekitoa