"Gio is an excellent player who unfortunately only hinted at his potential in his year here at Borussia, partly because a number of injuries meant he couldn't consistently show his ability on the pitch," said sporting director Schröder.
"We've said clearly that we want and need to change and shrink the squad.
"So we've jointly concluded that a split this summer is the right step for both sides."
Borussia Monchengladbach are set to receive up to €4m for the American, who signed a contract with the Ligue 1 club running until 2031.
The move ends a difficult single season in Germany for Reyna, who arrived from Borussia Dortmund last summer but never established himself amid recurring fitness problems, managing only a handful of appearances before losing his place to younger academy options as the campaign wore on.
At Strasbourg, he joins a club building around young attacking talent under BlueCo's wider ownership network, giving him a fresh platform to rebuild momentum after a stop-start period since leaving Borussia Dortmund.
Schröder's comments make clear the move is as much about squad management at Gladbach, who have been open about needing to trim numbers this summer, as it is about Reyna's individual struggles, framing the exit as a mutual decision rather than one forced by either side alone.