The 27-year-old, former Middlesbrough and Blackburn Rovers stopper is close to signing with the two clubs having agreed a deal.
Personal terms are now being discussed with the ex-England under-21 man, who has made more than 250 Championship appearances in his career - with a three-year deal expected to be announced imminently.
Steele, who still had three years left to run on his contract at Sunderland, will come in as back-up to number one Maty Ryan to solve the question marks over Albion’s goalkeeping situation next season.
It is understood that current number two Tim Krul and Niki Maenpaa will leave the club this summer in search of regular football.
Ryan is also likely to miss several matches in January due to Australia’s involvement in the Asian Cup in the United Arab Emirates.
Steele has had an up and down year at Sunderland after joining from Ewood Park on a Â£500,000 deal 12 months ago.
He was constantly in and out of the team, although did return in the latter part of the season which saw Sunderland suffer their second successive relegation.
Steele has drawn criticism from factions of fans on Wearside having lost his place in the team initially after just four games.
But Albion have spotted something in the goalkeeper and are keen to help him escape from the north east.
Meanwhile, Albion’s interest in Steele’s team-mate Paddy McNair has ended.
According to reports in the north east, the Seagulls made two bids for the Northern Ireland international, topping out at Â£3.75million in an attempt to bring him back to the Premier League.
That did not meet the Black Cats’ valuation and since a bid from Middlesbrough the 23-year-old has been accepted, although Stoke City could be about to make a late attempt to hijack the deal..