The 17-year-old has progressed through the youth system at the Conference side and, prior to his move, he had featured six times in the senior team this season, scoring his first goal for the club in a 2-2 draw with Tamworth.
Reid signed a new contract at Wrexham earlier this year, but Kevin Thelwell, Wolves' head of football development and recruitment, says the club have been monitoring the striker's progress for some time.
"Bradley is a player, similar to the signings of the young lads announced last week, who will go in with the Under-21 group when he arrives, but can also play at Under-18 level," Thelwell told Wolves' official website.
"He is actually a player who we nearly signed at Under-16 level after he'd been scouted by Alan Travis but unfortunately at that stage he suffered an injury.
"We have monitored his progress ever since and are delighted to have agreed a deal with Wrexham to bring him to the club where we are confident he will continue to develop."
Meanwhile, one player to exit Molineux is Austrian defender Georg Margreitter, who has joined FC Copenhagen on a season-long loan, linking up with former boss Stale Solbakken.