Rea returned to the club after suffering a serious knee injury during his loan spell with Wigan Athletic in March, and has since continued his rehabilitation at The Brache.
The 28-year-old, who has made over 200 appearances for the club, will remain with the Hatters in the short-term to continue full training and gain fitness in match action with the development squad.
Once ready to play first-team football again, he will join up with the Robins who are currently 18th in League One ahead of their away match against Bolton Wanderers on Saturday.
"This is a good move for Glen and the right thing for him as he continues his recovery from injury," said manager Rob Edwards. "He needs to be able to continue the great progress he's been making in recent weeks, and I've been so impressed with his work ethic and attitude around the club, but Glen recognises it's going to be hard for him to break into the team here so late in the season.
"He will only go there when he's ready, but it's a great opportunity for him to be able to get some competitive first team football at a good club in Cheltenham."
Good luck with the move when the time is right, Glen!