The 20-year-old centre-half, who recently agreed a new contract at White Hart Lane, has represented England at U19 and U20 levels.
"I'm thrilled to have signed Leigh on a season-long loan," Bees chief Andy Scott told the club's official website.
"He is a very talented player that I have been keeping an eye on for a couple of years now.
"He is an uncompromising centre-half who I'm sure will settle in well in League Two. He is desperate to do well, he wants to play games and gain experience of League football - he has a real hunger to succeed.
"I've had some very good reports from (Spurs coach) Alex Inglethorpe, Leigh is a player of good character, I've been very impressed by what I've seen and heard and I'm delighted to have him on board."