The former Rangers midfielder, who remains a popular figure with supporters, served as assistant under Rangers boss Ian Holloway last season.
The 40-year-old has also been a youth coach at Loftus Road, as well as spending time out in the MLS.
Bircham is desperate to land the job and has made it clear to club hierarchy that he is more than prepared to work under a strict playing budget, with Rangers set to cut further costs this summer with several first-team players out of contract.
talkSPORT.com understands Bircham has another enticing job offer on the table, but is waiting for the conclusion of this recruitment process before making a decision on that position.
Tim Sherwood remains director of football’s Les Ferdinand’s first choice, but the former Tottenham boss still has reservations over which direction the club is going in.
While Sherwood is keen to return to management – the last role he held was back in 2015 with Aston Villa – he knows his next move is do or die in terms of his prospects as a No 1.
Chris Ramsey would almost certainly come in as Sherwood’s assistant, should his expected talks with the club’s owners go smoothly.