The former Arsenal talent is out of contract in the summer.
The Norwegian international's current deal with Gladbach expires in the summer and despite a new deal being on offer he has been considering his future over the past few months.
West Ham would certainly be benefiting if they did bring him into the club.
Nordtveit is an excellent talent who can operate in a number of positions with ease. He is best suited to a defensive midfield role but started his career as a central defender and has also operated at right-back when needed.
He would add genuine depth and versatility to the West Ham squad - which can prove vital in the tougher stages of the Premier League season.