Mike Ross is Ireland's most serious doubt ahead of Saturday's first Test against New Zealand at Eden Park.
The Leinster prop suffered a slight hamstring tear in the RaboDirect Pro 12 final against Ospreys on May 27 and limped out of the game after just 16 minutes.
Ireland team manager Michael Kearney revealed Ross is facing an uphill battle to be fit in time to face the All Blacks but Cian Healy and Sean O'Brien, the team's two other injury concerns, are nearing full fitness.
He said: "Mike Ross is rehabbing from his hamstring tear. It is going well but he hasn't trained fully and remains in some doubt ahead of the weekend."
Kearney added: "Cian trained fully (on Monday) and is good to go.
"Sean O'Brien is continuing to rehabilitate well and he accelerated his rehab. We are hopeful that he will be available for selection at the weekend."
Ireland assistant coach Les Kiss believes New Zealand will be determined to re-establish Eden Park as a home fortress in their first run-out since they won the Rugby World Cup.
He said: "From the squad they picked it is going to be a fantastic series. We're playing the world champions, the best in the world bar none.
"We expect them to throw everything at us. They have a proud history at Eden Park.
"Taking on the All Blacks at any time is a tough prospect. I know that they are very proud at what they have achieved in Eden Park.
"I know Australia have spoken up in recent times about going there and breaking that there but [New Zealand] just keep turning up at that ground and nailing it."