Wilkinson has not played international rugby since he inspired England to World Cup glory in 2003, but could tour if he returns to form for club side Newcastle.
The 25-year-old wrote in The Times: "It is disappointing not to be named in the British Isles squad, but I can assure you I am doing and will do everything I can to make it to New Zealand.
"I have always had a lot of respect for Sir Clive Woodward and the way he has handled this with me only increases that; the fact that he has given me a chance and told me what he was thinking and the decision he was coming to and why.
"He explained to me what was required of me. It makes life much easier."
Newcastle have just two Zurich Premiership fixtures left to play, away to Northampton on Friday and at home to London Irish on April 30.
However, they could still qualify for the Zurich Wildcard play-offs and Wilkinson added: "I am not sure yet whether I will be playing on Friday.
"I have a few things to check out. I am pretty happy with where I am fitness wise. I am trying to be sensible and make good decisions."
Lions captain Brian O'Driscoll is backing Wilkinson to make the trip.
He told BBC Radio Five Live: "Hopefully he can get a couple of games under his belt and prove his fitness and I'm sure he'll be added to the squad before long."
Woodward explained his decision at Monday's press conference and added that he had every faith in the other three fly-halves selected - Stephen Jones, Ronan O'Gara and Charlie Hodgson.
"I think it would have been a mistake for me to put Jonny in the tour party until I'd seen him play," said Woodward.
"We're going to a very intense environment and I need to know he's back to something like his best.
"It would be ridiculous picking a player who hasn't played for such a long time until we'd actually seen him play.
"If he can come through it will be a bonus but, if he doesn't, I'm delighted with the three 10s I'm picking."