Wilkinson, whose knee injury in March was the latest in a succession of problems, was not named in the initial Lions squad to travel to New Zealand but has been given a chance to prove his fitness.
Woodward will continue to monitor Wilkinson's progress in the Premiership wildcard play-offs before deciding whether he should be included. England team-mates Tindall and Vickery are also hoping to join the squad.
Woodward said: "I have spoken to Jonny after his game this weekend and am delighted that he came through the game without any problems.
"The next 10 days are key not only for Wilkinson but for Tindall and Vickery to monitor their progress and so no decisions or announcements will be made before this time on their possible involvement in the Lions tour party."