The youngster, who was released by the Bees at the end of last season, has been brought in on trial by Coppell on the advice of ex-Brentford boss Wally Downes - now a coach at Reading Football Club.
McNamara, who is yet to make his football league debut, will be given two weeks to prove that he is good enough to make the grade at the Madejski Stadium. The versatile young player was used primarily on the wing last season, although he can play left-back and also up front.
McNamara was set to be handed a new deal by then-Brentford manager Wally Downes but the 43-year-old got the sack before the left-sided man got his new deal. When new man Martin Allen took control of the Griffin Park side he decided against keeping hold of the youngster.
Speaking to the official site, Downes explained the bizarre scenario: "When I was still manager at Brentford last season, we verbally agreed to take Steve on for another year, but the new manager Martin Allen then decided he didn't want him and he was released."
If Coppell decides to bring in McNamara on a permanent basis, Downes believes the left-footer will start life in the reserves before eventually progressing to first team football, as he considers the youngster to be a good player.
"He's certainly got something about him and I was genuinely very surprised when Brentford let him go. He's done well so far in training and he'll probably get a chance in one of the friendlies next week," Downes added.
Meanwhile, 22-year-old Sydney United goalkeeper Liam Reddy has also joined the Royals on trial. The Australian, who was recommended to the club by former player Andy Bernal, will spend pre-season with Reading as he attempts to earn himself a permanent deal.
Coppell is considering bringing in a new goalkeeper to act as cover for Marcus Hahnemann following the departure of Jamie Ashdown.