England star Wilkinson was hurt during the 48-18 second Test defeat against New Zealand in Wellington last Saturday, experiencing neck and shoulder problems after making a tackle.
Wilkinson sustained numbness in his arm as a result, continuing a catalogue of injury troubles which have troubled him since the 2003 World Cup.
Stephen Jones and Charlie Hodgson, who starts for the Lions against Auckland, are on stand-by to replace him in next Saturday's final Test.
But the Lions say that no decision has been made on Wilkinson's availability to face the All Blacks at Eden Park.
"Jonny continues to receive treatment, and continues to improve," said a Lions spokesperson.
"A decision on availability for selection will be made at the appropriate time."
The Lions have already lost Wilkinson's fellow World Cup stars Lawrence Dallaglio and Richard Hill through injuries, while captain Brian O'Driscoll's tour ended prematurely because of a dislocated shoulder.
Wilkinson had told The Times: "As much as it is bad to say that I am used to these, it is also true that I am used to recovering from them.
"That is why I have not ruled anything out. I will take it day by day and go by the physio's advice."
Wilkinson wants the Lions to leave New Zealand with pride restored by victory in the third Test.
"Whether I can play any part in achieving this I don't know. After the two knocks I sustained in the second international on Saturday, it could be a lot worse so I am actually quite relieved.
"When I woke yesterday (Sunday) morning, I felt better than I thought I would because after the match I was unable to move much at all. So the injury has improved with time."