Gardner was spotted at the Lilleshall Academy Exit Trials, after Spurs told him he had no future at White Hart Lane.
He performed well last Saturday in The Robins Under-19 side, playing as a centre-half in a 0-0 draw against West Ham.
Meanwhile, the Ashton Gate club have dealt a further injury blow, as midfield youngster Aaron Brown will have x-rays on his collarbone, with fears that he could be lost for the rest of the season.
Brown and team-mate Kevin Amankwaah were injured in car accident over the weekend, as the two were returning from a trip to Torquay.
City Academy director Tony Fawthrop told the club's official website: "X-rays were not taken on Aaron when he first went to hospital because there was a lot of bruising and swelling.
"Now it will be possible to get a better idea of his injuries. Even if his collarbone is not broken, he looks sure to miss several games and maybe the rest of the season."
Amankwaah suffered a broken neck but will make a full recovery, despite also being lost for the rest of the season.
He is facing an operation to be fitted for a 'halo', a frame that immobilises the neck and spine, while his injuries heal.
