The 28-year-old British and Irish Lion's full-back is on the lookout for a new club this summer having been told that he was not getting a new contract with French outfit Toulon.
Halfpenny had been linked with a return to Wales, but Scarlets and Cardiff Blues have been priced out of a move despite only having to pay a 40% share of his wages under the terms of a Welsh Rugby Union national dual contract.
That paved the way clear for Wasps to try and get the Wales international on board for the Aviva Premiership campaign, but after months of pursuing him, Young has revealed that he is off their wanted list.
"We were really keen to get Leigh on board and we tried over a period of months," he told The Rugby Paper.
"But we found it very hard to get a decision out of him and in the end we couldn't get one. We have moved on since then."
Reports suggest that Wasps' Premiership rivals Bath are now the favourites to sign Halfpenny.