Irish hooker Shane Byrne has also remained behind after taking a knock during the same training session.
Both players had been due to take part in community activities in and around Rotorua that have been arranged by the Lions.
Wilkinson was due to visit the flood victims of rural village Matata, 40 minutes outside Rotorua, along with seven other players, team manager Bill Beaumont and defence coach Mike Ford.
Byrne had been due to visit Pukekohe for the unveiling of a Wall of Legends at the Counties Manukau Rugby Union.
Wilkinson, who started his first international in 18 months against Argentina in Cardiff, was not included in the squad to face Bay of Plenty.
But he now seems certain to be involved in the Taranaki game, though whether he will start the game remains to be seen.
Stephen Jones has now arrived in New Zealand and with Ronan O'Gara starting the Bay of Plenty game, the Welshman and Charlie Hodgson will be pressing to start against Taranaki.