The midfielder’s contract runs out at the end of the season and the Bundesliga side said in a statement on Sunday that a move to England has been agreed.
“After five and a half years with Borussia Monchengladbach, Havard Nordtveit will leave the club in the summer,” the statement read.
“The Norway international has chosen not to extend his contract and will move to England at the end of the season.”
A return to North London has also been reported, but it is believed the player wants regular first-team football, which indicates a move to Slaven Bilic’s Hammers seems more likely.
The 25-year-old can play in defence or central midfield and has been an integral part of Die Borussen’s side as they qualified for Europe in three of the past four seasons.
Since joining in 2010, he has made 229 first-team appearances and scored 13 goals, including one in a victory against Klopp’s Dortmund last season.
He has also been capped 24 times for Norway, scoring twice.