The 31-year-old also played Southend United, but is seeking to earn a deal at Broadfield Stadium for next season.
The 31-year-old was involved in Saturday's match against Fulham, and will continue his trial in the Reds' next game against Brighton later this week.
To those of a certain generation of Spurs fans, Barnard was a promising individual that looked to be the first homegrown striker in many years to break into the first team and make an impact in the Premier League.
From a club who existed on a diet of Rory Allen and Neale Fenn during the 90s, this wasn't saying much of course, and so it turned out that Barnard didn't quite make the grade at White Hart Lane, leaving after coming through the academy system, and making 3 appearances for the first team between 2006 and 2007.
He ended up at Southend United though, which is where he really found his feet, scoring enough goals for the Shrimpers to convince Southampton, then in League One, and he scored plenty more goals for the Saints as Alan Pardew and then Nigel Adkins helped the South Coast club back into the Championship.
But Barnard again didn't really get a much of a chance once the club had got promoted, and the striker took in a variety of loans before his contract expired in 2014, with varying levels of success.
Last season he spent his third spell with Southend, but, struggling for game time and goals when he was on the pitch, he found himself on loan to Stevenage in the middle of the campaign, and unable to get fit once he returned to Roots Hall.
Now 31, Barnard could now be given a contract at Broadfield Stadium, knowing that with Crawley only being relegated from League One last season, he has a good chance of promotion in the coming campaign if he is signed up.