He is yet to train with England, but he has been selected to travel on the initial flight because Lancaster has not ruled out picking him for the first Test against the All Blacks on June 7.
Lancaster said: "Billy (Twelvetrees) is going well. But we're optimistic enough to take him on the initial flight with a view to assessing him when we get out there.
"He's really going to have to be training fully at the start of that Test week in order to be considered.
"But he's certainly made enough progress to be under consideration for the first Test and definitely for the second, so that's why he's on the plane."