Jones belatedly arrived in New Zealand on Tuesday after helping his French club side, Clermont Auvergne, qualify for next season's European Cup.
When the initial 44-man squad was announced, Sir Clive Woodward made it clear Jones was the first choice fly-half. Wilkinson, at that point, still had to prove his fitness and his form.
But while Jones was finishing off his club season, Wilkinson was added to the Lions squad and then made an impressive return to international rugby against Argentina.
Despite Wilkinson's impressive comeback, the 27-year-old Jones did not hold back on his desire to become the latest in a long line of Welsh Lions fly-halves.
"It's been a very enjoyable year and I feel very proud to achieve the Six Nations Grand Slam and to finally get selected for the Lions is an honour.
"If you were to get that Test jersey it would be a dream," said Jones.
"I am definitely not going head-to-head (with Wilkinson). We are all here as British & Irish Lions. We want this to be a successful tour and be part of a successful team.
"Whoever gets selected will have the backing of those not involved.
"What's great is that all the players are available in the number 10 position. Everybody will have an opportunity to play and whoever makes the most of that will be given the opportunity to play.
"Everybody in the squad has that mindset that they want to play in the Tests but there is so much competition it's incredible."