The Bundesliga outfit made the announcement after the 25-year-old rejected their offer of a new contract.
"Havard Nordtveit has opted not to extend his contract and will move to England in the summer," the club posted on their official Twitter page.
The Norwegian was on Arsenal's books as a trainee and had been linked with a return to the Emirates Stadium.
However, the Daily Mail recently claimed that West Ham United had beaten the Gunners to the defensive midfielder's signature.
Nordtveit has made over 200 appearances for Monchengladbach since 2010.