The 33-year-old Toulon fly half was omitted from the 37-man group with suggestions that head coach Warren Gatland did not want to use any players plying their trade with French clubs, as they could still be playing when the tour begins on June 1.
Wilkinson has said that he would be happy to be part of the team if it was possible but admitted that his club side would take precedence over any potential inclusion.
"I'll see how I am at the end of the club season and if Warren needs me and I am able to get involved, that would be fabulous," Wilkinson told The Times.
"I told him where I am at the moment, that there's a bit of day-to-day management where my body is concerned. I was phenomenally proud to be considered for the Lions, so it's very disappointing that I'm unavailable."
"If Toulon manage to get through to the final of the Top 14 [league on June 1], my absolute priority lies with them and helping to get the best out of a team that has been so good to me for so long.
"It is, after all, these guys who have got me to the stage where I could even be considered for the Lions."
The tour Down Under begins for Gatland's side against the Barbarians in Hong Kong, the same day as the French Top 14 final at Stade de France.