The 37-year-old, whose deal expires in the summer, is Albion's joint leading scorer this season and is the oldest outfield player in the Premier League.
"He has been fantastic and we are very close to agreeing a new deal," said Baggies boss Tony Pulis.
"He has continuously put in performances which are special for someone of his age."
The Northern Ireland international, who did not make his professional debut until he had turned 24, has made 205 appearances since joining Albion from Ipswich in the summer of 2011.
Pulis added: "You shouldn't use the word 'freak' but Gareth's fitness levels and the way he looks after his life on and off the pitch are first class.
"He is a wonderful lad, a wonderful character and he has had a great season.
"There is no reason why he can't carry on for another year for us."
West Brom centre-back Gareth McAuley is close to signing a new contract, keeping him at the club next season.