The former Luton centre-half moved to Carrow Road in August and the two clubs have agreed in principle a deal, for an undisclosed fee, which will see Barnett join City on a three-and-a-half-year contract.
He joined Albion from Luton for an initial £2.5million fee in July 2007 and his Hawthorns contract is set to expire in June.
Norwich manager Paul Lambert said: "Leon has been colossal for us and has become a fans` favourite.
"I don`t think anyone can say that he hasn`t given everything he`s got and I`m delighted he will join us permanently."
Source: PA
Source: PA