The former Arsenal trainee has been linked with a return to the Emirates Stadium as well as a move to Liverpool and West Ham, having shown good form for the Bundesliga outfit.
And the club confirmed via their Twitter account that he will depart at the end of the season, saying: "Havard Nordtveit has opted not to extend his contract and will move to England in the summer."
Nordtveit, who can also play as a central defender, will be out of contract at the end of the current campaign and Gladbach had been hoping to extend his deal, with sporting director Max Eberl saying last month: "We have had a great time together and we would like to extend that. Now he has to decide."
The 25-year-old Norway international has scored four goals in 31 appearances in all competitions for the German club since joining from Arsenal in 2011.