The 33-year old comes out of contract at the end of next season but his fellow Northern Irishman Brunt has urged Baggies officials to tie him up longer than that.
McAuley joined Albion from Ipswich Town in 2011 and has emerged as a key player for the West Midlands club this season.
"I can't speak highly enough of him as a lad, or of his performances on the pitch," Brunt told the Express & Star.
"He's a good friend of mine as well so I would love him to stay, and hopefully the club will sort it out.
"He's been massive for us. From day one, since he stepped into the team, he has been our most consistent player.
"And that's coming from someone who had never played in the Premier League until he was 31.
"I had known him for a long time and I knew he was a good defender. But when it came to the Premier League it was as unknown for me as it was for everybody else, and probably to him as well.
"But he has stepped up and been fantastic for us."