The 21-year-old has been widely linked with a move away from the Bundesliga outfit, with Barcelona reported to be leading the chase for his signature as the club seeks to replace Victor Valdes.
Ter Stegen has a contract at Borussia Park until 2015, but attempts to get him to sign a new deal have been rejected, according to sporting director Max Eberl.
The club announced the news from Eberl on its official Twitter account, posting: "Ter Stegen will leave Borussia. Marc has informed us that he will not accept our offer."
Ter Stegen has kept six clean sheets for Monchengladbach this season, helping the team climb to third in the Bundesliga.