The Honduras international has been a long-term target for Arsenal.
"We are pleased to finalise Andy's stay with Tottenham," DCU GM Dave Kasper commented on the club's official website.
"I think this experience will be tremendous for him. He will come away from this as a better player, and we look forward to having him back with us shortly.
"It's great for our players to get into a different environment and have a different experience. We feel it's good for their continued development.
"It's certainly aligned with the philosophy that Jurgen Klinsmann has made public, which is that our offseason is too long and our players need to continue training."