He played for the Baggies just 12 times last season, having joined from Ipswich back in 2011.
Rangers manager Steven Gerrard was asked about McAuley after the 1-0 Old Firm defeat to Celtic on Sunday.
He said: "There is no rush. It will be the last deal."
Gerrard has already revolutionised Rangers during his short stint at the club, signing 14 players this summer.
Rangers news: Gareth McAuley has signed for the Glasgow club (Image: GETTY)
Rangers news: Gareth McAuley (left) was playing for West Brom last season (Image: GETTY)
Rangers news: Northern Ireland's Gareth McAuley has supposedly signed a one-year contract (Image: GETTY)
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The Ibrox side lost their first match of the season in the 1-0 defeat at Celtic Park and sit seventh in the Scottish Premiership after four games.
Sky Sports News claim the deal for McAuley has been wrapped up.
McAuley revealed last week that he had a number of offers on the table.
“I’m lucky that I’m able to be picky,” he said. “There are lads who go through this sort of thing season after season, waiting for clubs, moving around all over the place, but I’m not thinking ‘I have to pay the bills’.”