The 21-year-old New Zealander – a former junior Otago Highlander – was signed on a short term contract as cover for Bath's World Cup backs but is being released as the club can't afford to keep him.
"He's going to Saracens because we couldn't fit him under the salary cap, but he's been great for us," head coach John Connolly told BBC Radio Bristol.
"We would have liked to have kept him because we think he's got a lot to offer."
Kydd is EU qualified and has played representative rugby for New Zealand at schoolboy, U19 and U21 level as well as being a part of the NZ Universities side last season.
Meanwhile, Saracens centre Kevin Sorrell has been ruled out for a further four weeks after fracturing his hand in training yesterday on Wednesday.
Sorrell had been making good progress after suffering a groin injury against NEC Harlequins in October but suffered the latest setback in his first light training session with the rest of the squad.