The 19-year-old won the Aviva Premiership A League title with the Tigers, while he made his senior debut in the LV= Cup.
However Markham, who came through the academy set-up at Sale before moving to Welford Road, is relishing the chance to develop his game at Gloucester.
"Gloucester is a great club in a great rugby community and it's a chance for me to explore the game more and test myself against some of the best players in the country," he told the club's official site.
"It's about developing myself as a person and as a player. Gloucester have shown in the past that they're not afraid to give young players a chance and hopefully I'll get some game time as the season goes on.
"I'm really looking forward to meeting the rest of the squad and putting in the hard work in training but especially working with guys like Alex Brown, Jim Hamilton and Will James and learning from them, developing my game and making the most of any opportunities that come my way."
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