The ex-midfielder, who began his career at Wolves before going on to feature for West Brom during a brief stint in 2012 - has most recently been a coach at The Bees under Thomas Frank.
Frank departed the club last week to join Tottenham Hotpsur, and over the weekend Andrews has emerged as a serious contender to replace him.
Andrews has been Brentford's set piece coach, after moving into coaching following a stint working as a pundit for Sky Sports.
According to reports, he is well regarded at the Premier League club.
Brentford have already promoted from within in the past - with Frank having served as number two to former boss Dean Smith during his time at the club.
According to Sky Sports, the club are yet to make a final decision on the job, but Andrews is one of a handful of names who are being seriously considered to replace Frank.